Trading, Profits, and Volatility in a Dynamic
Information Network Model∗
Johan Walden†
November 14, 2013
Abstract
We introduce a dynamic noisy rational expectations model, in which information diffuses
through a general network of agents. In equilibrium, agents’ trading behavior and profits
are determined by their position in the network. Agents who are more closely connected
have more similar period-by-period trades, and an agent’s profitability is determined by a
centrality measure that is closely related to so-called Katz centrality. The model generates
rich dynamics of aggregate trading volume and volatility, beyond what can be generated
by heterogeneous preferences in a symmetric information setting. An initial empirical investigation suggests that price and volume dynamics of small stocks may be especially well
explained by such asymmetric information diffusion. The model could potentially be used
to study individual investor behavior and performance, and to analyze endogenous network
formation in financial markets.
Keywords: Information diffusion, information networks, heterogeneous investors, portfolio
choice, asset pricing.
∗
I thank seminar participants at the Duisenberg School of Finance, Maastrict University, the Oxford-Man
Institute, Rochester (Simon School), Stockholm School of Economics, and USC, Bradyn Breon-Drish, Laurent
Calvet, Nicolae Gârleanu, Brett Green, Pierre de la Harpe, Philipp Illeditsch, Matt Jackson, Svante Jansson, Ron
Kaniel, Anders Karlsson, Martin Lettau, Dmitry Livdan, Hanno Lustig, Gustavo Manso, Christine Parlour, Han
Ozsoylev, Elvira Sojli, Matas Šileikis, and Deniz Yavuz for valuable comments and suggestions.
†
Haas School of Business, University of California at Berkeley, 545 Student Services Building #1900, CA
94720-1900. E-mail: [email protected], Phone: +1-510-643-0547. Fax: +1-510-643-1420.
1
Introduction
There is extensive evidence that heterogeneous and decentralized information diffusion influences
investors’ trading behavior. Shiller and Pound (1989) survey institutional investors in the NYSE,
and find that a majority attribute their most recent trades to discussions with peers. Ivković and
Weisbenner (2007) find similar evidence for households. Hong, Kubik, and Stein (2004) find that
fund managers’ portfolio choices are influenced by word-of-mouth communication. Heimer and
Simon (2012) find similar influence from on-line communication between retail foreign exchange
traders.
Such information diffusion may help explain several fundamental stylized facts of stock markets. First, investors are known to hold vastly different portfolios, in contrast to the prediction
of classical models that everyone should hold the market portfolio. The standard explanations
for such diverse portfolio holdings are hedging motives and heterogeneous preferences, but several studies indicate that there are limitations to how well such motives can explain observed
heterogeneous investor behavior.1 With decentralized information diffusion, however, it is unsurprising if significantly heterogeneous behavior of investors is observed in the market. Second,
stock markets are known to experience large price movements that are unrelated to public news,
as documented in Cutler, Poterba, and Summers (1989), and Fair (2002). These studies find
that over two thirds of major stock market movements cannot be attributed to public news
events, suggesting that there are other channels through which information is incorporated into
asset prices. Third, the dynamics of trading volume and asset prices are known to be very rich.
Returns and trading volume in many markets are heavy-tailed, time varying, show “long memory,” and are related to each other in a complex way (see Gabaix, Gopikrishnan, Plerou, and
Stanley 2003; Karpoff 1987; Gallant, Rossi, and Tauchen 1992; Bollerslev and Jubinski (1999);
Lobato and Velasco 2000). Lumpy information diffusion provides a potential explanation for
such behavior. In periods when more information diffuses, volatility is higher, as is trading volume (see, Clark 1973; Epps and Epps 1976; Andersen 1996). To generate rich trading volume
dynamics, however, such information diffusion must necessarily be heterogeneous.
In this paper, we follow a recent strand of literature that uses information networks to model
information diffusion (see Colla and Mele 2010, Ozsoylev and Walden 2011, Han and Yang 2013,
and Ozsoylev, Walden, Yavuz, and Bildik 2013). Agents who are directly linked in a network
share information, which consequently diffuses among the population over time in a well-specified
manner. This literature has made important observations about effects of information networks,
but several key questions remain open. How does the network structure in a market determine
the dynamic trading behavior of its agents, and their performance? How does the network
structure influence aggregate properties of the market, e.g., price volatility and trading volume?
What does heterogeneous information diffusion “add” compared with, e.g., what can can be
1
For example, Massa and Simonov (2006) find that hedging motives for human capital risk—a fundamental
source of individual investor risk—does not explain heterogeneous investment behavior among individual investors well. Similarly, Calvet, Campbell, and Sodini (2007) and Calvet, Campbell, and Sodini (2009) find that
diversification and portfolio rebalancing motives do not explain investors’ portfolio holdings well.
1
generated by heterogeneous preferences alone? These questions are obviously fundamental for
our understanding of the impact of information networks on financial markets, and ultimately
for how well such information networks can explain observed investor and market behavior. In
this paper we analyze these questions.
We introduce a dynamic noisy rational expectations model in which agents in a network
share information with their neighbors. Agents receive private noisy signals about the unknown
value of an asset in stochastic supply, and trade in a market over multiple time periods. In each
period, they share all the information they have received up until that point with their direct
neighbors, leading to gradual diffusion of the private signals. The structure of the network is
completely general.
As a first contribution, we prove the existence of a noisy rational expectations equilibrium,
and present closed-form expressions for all variables of interest. Theorem 1 provides the main
existence and characterization result for a Walrasian equilibrium in a large network economy. To
define the large economy equilibrium, we use the concept of replica networks, assuming that that
there is a local network structure (e.g., at the level of a municipality) and that there are many
similar such local network structures in the economy. This allows for a clean characterization of
equilibrium, as well as justifies the assumption that agents act as price takers and are willing to
share information. We know of no other network model of information diffusion in a centralized
financial market (i.e., exchange) that allows for a complete characterization of equilibrium, that
is completely general with respect to network structure, and that is based on first principles of
financial economics.
The structure of the network is crucial in determining asset pricing dynamics. We show in
a simple example that price informativeness and volatility at any given point in time does not
only depend on the specific information agents in the model have obtained at that point, but
also on how the information has diffused through the network. The equilibrium outcome thus
depends on complex properties of the network, beyond the mere precision of agents’ signals at
any specific point in time.
We next study how the network structure determines the trading behavior and profitability
of agents. A priori, one may expect that portfolio holdings of agents who are close in the
network should be positively correlated, whereas their trades may be negatively correlated in
some periods, because some agents trade earlier on information than others and may be ramping
down their investments to realize profits when other agents are ramping up their investments.
We show that, contrary to this intuition, the period-by-period trades of agents in the network are
always positively related, and increasingly so in the degree of the overlap in their connectedness.
This result justifies the use of data on trades to draw inferences about network structure, as has
been done in previous studies, providing our second contribution.
As a third contribution, we study what determines who in the network makes profits. It
is argued in Ozsoylev and Walden (2011) and elsewhere that some type of centrality measure
should determine agent profitability.2 Centrality—a fundamental concept in network theory—
2
In an empirical study, Ozsoylev, Walden, Yavuz, and Bildik (2013) study the trades of all investors on the
2
captures the concept that it is not only who your direct neighbors are that matters, but also who
your neighbors’ neighbors are, who your neighbors’ neighbors’ neighbors are, etc. The argument
is that agents who are centrally placed tend to receive information signals early, and therefore
perform better in the market than peripheral agents, who tend to receive information later. It
is not a priori clear, however, which centrality measure—among several—that is appropriate in
this context.
We show that profitability is determined by a centrality measure related to eigenvector centrality or, more generally, to so-called Katz centrality. To the best of our knowledge, this is the
first complete characterization of the relationship between agent centrality and performance in
a general information network model of financial markets. For large random networks, Katz
centrality dominates other common centrality measures in our model, i.e., it dominates degree
centrality, average distance and closeness centrality. We show in simulations that the ranking
of centrality measures also holds in medium-sized random networks. Thus, the use of centrality measures when studying individual investor behavior (e.g., in Ozsoylev, Walden, Yavuz,
and Bildik 2013) is theoretically justified. The result is quite intuitive: The Katz centrality
measure provides the best balance between direct connections and connections at farther distances, whereas degree centrality focuses exclusively on direct connections, and average distance
and closeness centrality are tilted toward connections at far distances. Our main results that
characterize trading behavior, profitability and welfare of agents, and the relationship between
centrality and profitability, are Theorems 2-4.
Our fourth contribution is to derive and analyze several aggregate results regarding the
dynamic behavior of price volatility and trading volume in the model. Specifically, the network
structure in an economy is an important determinant of volatility and trading volume in the
time series, as well as of the relationship between the two. Several stylized properties naturally
arise in the model, for example, persistence of shocks to volatility and trading volume, as well
as lead-lag relationships between the two.
We show that the rich dynamics of volatility and volume in our general information network
model cannot be generated by heterogeneous preferences alone in a symmetric network. Again,
the intuition is straightforward: With symmetry, there can be no bottlenecks in the information
network. This in turn implies that the rate of information diffusion over time must be unimodal,
i.e., it cannot slow down and then speed up again. Consequently, aggregate market dynamics
are restricted under symmetry, whereas (as we show) there are no such unimodality restrictions
in the general case.
Finally, we study volatility and volume of a sample of U.S. stocks. We compare Large Cap
and Small Cap companies, and find that the behavior of Small Cap companies is better explained
by asymmetric networks with a large degree of nonpublic information diffusion, suggesting that
information networks may be especially important for such stocks. Our theoretical results on
Istanbul Stock Exchange in 2005, and find a positive relationship between investors’ so-called eigenvector centrality
and profitability, but this choice of centrality measure is not theoretically justified. Several other finance papers
discuss and use various centrality measures without a complete theoretical justification, see e.g., Das and Sisk
(2005), Adamic, Brunetti, Harris, and Kirilenko (2010), Li and Schurhoff (2012) and Buraschi and Porchia (2012).
3
aggregate volume and volatility are summarized in Theorems 5-7.
The rest of the paper is organized as follows. In the next section, we discuss related literature.
In Section 3, we introduce the model and characterize equilibrium. In Section 4, we analyze
trading behavior and profitability of individual agents. In Section 5, we study the implications
of network structure for aggregate volatility and trading volume. Finally, Section 6 concludes.
All proofs are delegated to the appendix.
2
Related Literature
Our paper is most closely related to the recent strand of literature that studies the effects of
information diffusion on trading and asset prices. Colla and Mele (2010) show that the correlation of trades among agents in a network varies with distance, so that close agents naturally
have positively correlated trades, whereas the correlation may be negative between agents who
are far apart. Their model is dynamic, and assumes a very specific symmetric network structure, namely a circle, where each agent has exactly two neighbors. This restricts the type of
dynamics that can arise in their model. Ozsoylev and Walden (2011) introduce a static rational
expectations model that allows for general network structures and study, among other things,
how price volatility varies with network structure. Their model is not appropriate for studying
dynamic information diffusion, however, and is therefore not well-suited for several of the questions analyzed in this paper, e.g., the relationship between agent profitability and centrality,
and the short-term correlation between agents’ trade.
Han and Yang (2013) study the effects of information diffusion on information acquisition.
They show that in equilibrium, information diffusion may reduce the amount of aggregate information acquisition, and therefore also the informational efficiency and liquidity in the market.
Their model is also static, and does thereby not allow for dynamic effects. In an empirical
study, Ozsoylev, Walden, Yavuz, and Bildik (2013) test the relationship between centrality—
constructed from the realized trades of all investors in the market—and profitability. They find
that more central agents, as measured by eigenvector centrality, are more profitable. However,
they do not justify this choice of centrality measure theoretically. Pareek (2012) studies how
information networks—proxied by the commonality in stock holdings–among mutual is related
to return momentum.
A different strand of literature studies information diffusion through so-called information
percolation (Duffie and Manso 2007, Duffie, Malamud, and Manso 2009). In the original setting, a large number of agents meet randomly in a bilateral decentralized (OTC) market and
share information, and the distribution of beliefs over time can then be strongly characterized.
Recently, the model has been adapted to centralized markets, with exchange traded assets and
observable prices—a setting more closely related to ours. For example, Andrei (2012), shows
that persistent price volatility can arise in such a model. In contrast to our model, in which
some agents may be better positioned than others, these models are ex ante symmetric in that
all agents have the same chance of meeting and sharing information.
4
Babus and Kondor (2013) also introduce a model of information diffusion in a bilateral OTC
market. As in our paper, their network can be perfectly general. In contrast to our model, there
is no centralized information aggregation mechanism in their setting, and therefore no interplay
between diffusion through public and private channels. Moreover, agents have private values in
their model, and their model is static.
This paper is also related to the literature on information diffusion and trading volume
(Clark 1973). Lumpy information diffusion was suggested to explain heavy-tailed unconditional
volatility of asset prices, as an alternative to the stable Paretian hypothesis. Under the Mixture
of Distributions Hypothesis (MDH), lumpiness in the arrival of information lead to variation in
return volatility and trading volume, as well as a positive relation between the two (see Epps
and Epps 1976 and Andersen 1996). Foster and Viswanathan (1995) build upon this intuition to
develop a model with endogenous information acquisition, leading to a positive autocorrelation
of trading volume over time. Similar results arise in He and Wang (1995), in a model where an
infinite number of ex ante identical agents receive noisy signals about an asset’s fundamental
value. Admati and Pfleiderer (1988) explain U-shaped intra-daily trading volume in a model
with endogenous information acquisition.
Our paper further explores the richness of the dynamics of volatility and volume that arises
when agents share their signals, allowing for completely general asymmetry in how some agents
are better positioned than others. This extension may potentially shed further light on the
very rich dynamics of volatility and volume, and the relationship between the two (see Karpoff
1987, Gallant, Rossi, and Tauchen 1992, Bollerslev and Jubinski 1999, Lobato and Velasco 2000,
and references therein). A related strand of literature explores the role of trading volume in
providing further information to investors about the market, see Blume, Easley, and O’Hara
(1994), Schneider (2009), and Breon-Drish (2010). Our model does not explore this potential
informational role of trading volume.
Our study is related to the large literatures on games on networks, see the survey of Jackson
and Zenou (2012). The games in these models are typically not directly adaptable to a finance
setting. Our existence result and the characterization of equilibrium in a model based on first
principles of financial economics are therefore of interest. Since the welfare of agents in equilibrium can be simply characterized, our model could potentially also be used to study endogenous
network formation, see Jackson (2005) for a survey of this literature.
Finally, our paper is related to the (vast) general literature on asset pricing with heterogeneous information (see, e.g., the seminal papers by Grossman 1976, Hellwig 1980, Kyle 1985,
and Glosten and Milgrom 1985). Technically, we build upon the model in Vives (1995), who introduces a multi-period noisy rational expectations model in a similar spirit as the static model
in Hellwig (1980). Like Vives, we assume the presence of a risk-neutral competitive market
maker, to facilitate the analysis in a dynamic setting. This simplifies the characterization of
equilibrium considerably. The cost of this assumption is that asset prices simply reflect the
expected terminal value of the asset conditioned on public information at all points in time.
Since our focus is on volatility and trading volume, this is a marginal cost for us. Unlike Vives,
5
we allow for information diffusion among agents, through general network structures.
3
Model
There are N agents, enumerated by a ∈ N = {1, . . . , N }, in a T + 1-period economy, t =
0, . . . , T + 1, where T ≥ 2. We define T = {1, . . . , T }. Each agent, a, maximizes expected
utility of terminal wealth, and has constant absolute risk aversion (CARA) preferences with risk
aversion coefficient γa , a = 1, . . . , N ,
Ua = E[−e−γa Wa,T +1 ].
We summarize agents’ risk aversion coefficients in the N -vector Γ = (γ1 , . . . , γN ).
There is one asset with terminal value v = v̄ + η, where η ∼ N (0, σv2 ), i.e., the value is
normally distributed with mean v̄ and variance σv2 . Here, v̄ is known by all agents, whereas η is
unobservable.
Agents are connected in a network, represented by a graph G = (N , E). The relation E ⊂ N ×
N describes which agents (vertices, nodes) are connected in the network. Specifically, (a, a ) ∈ E,
if and only if there is a connection (edge, link) between agent a and a . We will subsequently
assume that there are many identical “replica” copies of this network in the economy, each copy
representing a “local” network structure. This will make the economy “large” and justify price
taking behavior of agents, as well as simplify the characterization of equilibrium. For the time
being, we focus on one representative copy of this large network.
We use the convention that each agent is connected to himself, (a, a) ∈ E for all a ∈ N ,
i.e., E is reflexive. We also assume that connections are bidirectional, i.e., that E is symmetric.
A convenient representation of the network is by the adjacency matrix E ∈ {0, 1}N ×N , with
(E)aa = 1 if (a, a ) ∈ E and (E)aa = 0 otherwise.
The distance function D(a, a ) defines the number of edges in the shortest path between
agents a and a . We use the conventions that D(a, a) = 0, and that D(a, a ) = ∞ whenever there
is no path between a and a . The set of direct neighbors to agent a is Sa,1 = {a : (a, a ) ∈ E}.
Moreover, the set of agents at distance m > 1 from agent a is Sa,m = {a : D(a, a ) = m},
and the set of agents at distance not further away than m is Ra,m = ∪m
j=1 Sa,m . The number of
agents at a distance not further away than m from agent a is Va,m = |Ra,m |. Here, Va,1 is the
degree of agent a, which we also refer to as agent a’s connectedness, whereas Va.m is agent a’s
mth order degree. We use the convention that Va,0 = 0 for all a. We also define ΔVa,m = |Sa,m |.
We define N -vectors V m , m = 1, . . . , N , where the ath element of V m is Va,m Equivalently,
Definition 1 The mth order degree vector, V m ∈ RN
+ , m = 1, 2, . . . , is defined as
V m = χ(E m )1.
(1)
Here E m is the mth power of the adjacency matrix, and χ : RN ×N → {0, 1}N ×N is a matrix
6
indicator function, such that (χ(A))i,j = 0 if Ai,j = 0 and (χ(A))i,j = 1 otherwise. Moreover, 1
is an N -vector of ones.
First order degree is commonly referred to as degree centrality.
Finally, the number of agents within a distance of m from both agents a and a is Va,a ,m =
|Ra,m ∩Ra ,m |, and the number of neighbors at distance exactly m from both agents is ΔVa,a ,m =
|Sa,m ∩ Sa ,m |
3.1
Information diffusion
At t = 0, each agent receives a noisy signal about the asset’s value, sa = v + σξa , where
ξa ∼ N (0, 1) are jointly independent across agents, and independent of v. At T + 1, the true
value of the asset, v, is revealed. It will be convenient to use the precisions τv = σv−2 and
τ = σ −2 .
The graph, G, determines how agents share information with each other. Specifically, at t+1,
agent a shares all signals he has received up until t with all his neighbors. We let Ia,t denote
the information set that agent a has received up until t, either directly or via his network.
It is natural to ask why agents would voluntarily reveal valuable information to their neighbors. Of course, in a large economy with an infinite number of agents, sharing signals with ones’
(finite number of) neighbors has no cost, since the actions of a finite number of agents will not
influence prices. Even with an economy of finite size, as long as signals can be verified ex post,
truthful revelation may be optimal in a repeated game setting, since an agent who provides
misinformation can be punished by his neighbors, e.g., by being excluded from the network in
the future. Even if signals are not ex post verifiable, it may still be possible for an agent to draw
inferences about the truthfulness of another agent’s signal, by comparing it with other received
signals. Again, the threat of future exclusion from the network, could be used to enforce truthful
information sharing. We therefore take the truthful information sharing behavior of agents as
given. A potentially fruitful area for future research is to better understand in which finite sized
financial networks truthful signal sharing can be sustained.
As in Ozsoylev, Walden, Yavuz, and Bildik (2013), we formalize the information sharing role
of the network by defining
Definition 2 The graph G represents an information network over the signal structure {sa }a ,
if for all agents a ∈ N , a ∈ N and times t = 1, . . . , T , sa ∈ Ia,t if and only if D(a, a ) ≤ t.
The information about the asset’s value that an agent has received through the network up until
time t can be summarized (as we shall see) by the sufficient statistic
def
za,t =
1 sj = v + ζa,t ,
Va,t
j∈Ra,t
7
V
3 The number of signals agent a receives at t is ΔV ,
where ζa,t = σa,t
a,t
2 ξa,t , and ξa,t ∼ N (0, 1).
and we therefore expect {ΔVa,t }a∈N ,t∈T to be important for the dynamics of the economy.
3.2
Market
The market is open between t = 1 and T + 1. Agents in the information network submit limit
orders, and a risk-neutral competitive market maker sets the price such that at each point in
time it reflects all publicly available information, pt = Et [v|Itp ], where Itp is the time-t publicly
available information set. At T + 1, the asset’s value is revealed so pT +1 = v. Before trading
begins, the price is set as the asset’s ex ante expected value, p0 = v̄.
To avoid fully revealing prices, we make the standard assumption of stochastic supply of
the asset. Specifically, in period t, noise traders submit market orders of ut per trader in the
network, where ut ∼ N (0, σu2 ). In other words, the noise trader demand is defined relative to
the size of the population in the information network. As argued elsewhere in the literature, the
noise trader assumption need not be taken literally, but is rather a reduced-form representation
of unmodeled supply shocks. It could, e.g., represent hedging demand among investors due to
unobservable wealth shocks, or other unexpected liquidity shocks. We do not further elaborate
on the sources of these shocks. We will use the precision τu = σu−2 .
Agents in the network are price takers. At each point in time they submit limit orders
to optimize their expected utility of terminal wealth. They thus condition their demand on
contemporaneous public information, as well as on their private information. An agent’s total
demand for the asset at time t is
xa,t = arg max E e−γa Wa,T +1 |Īa,t ,
(2)
x
subject to the budget constraint
Wa,t+1 = Wa,t + xa,t (pt+1 − pt ),
t = 1, . . . , T,
and his net time-t demand is Δxa,t = xa,t − xa,t−1 , with the convention that xa,0 = 0 for all
agents. Here, Īa,t contains all public and private information available to agent a at time t.
In the linear equilibrium we study, za,t and pt are jointly sufficient statistics for an agent’s
information set, Īa,t = {za,t , pt }, leading to the functional form xa,t = xa,t (za,t , pt ). Of course, an
agent’s optimal time-t strategy in (2) depends on the (optimal) future strategy. The dynamic
problem can therefore be solved by backward induction. The primitives of the economy are
summarized by the tuple M = (G, Γ, τ, τu , τv , v̄, T ).
We note that the assumption that the asset’s value is revealed at T + 1 means that any
residual uncertainty at T of the asset’s value is completely mitigated by T + 1. We think of
this as public information, which becomes available to all agents at T + 1. Alternatively, we
3
A variation is to let agents receive new private signals in each time period. The analysis in this case is
qualitatively similar, but not as clean because of the increased number of signals.
8
could have assumed that residual uncertainty is gradually incorporated into the market between
T and T for some T > T + 1, keeping the assumption that the information diffusion between
agents in the network only occurs until T .
The graph, G, determines how information diffuses in the network over time, whereas Γ
captures agent preferences. We wish to separate dynamics that can be generated solely by
heterogeneity in preferences from those that require heterogeneity in network structure. To
this end, we define an economy to be preference symmetric if γa = γ for all agents, and some
constant γ > 0. There are several symmetry concepts for graphs. The notion we use is so-called
distance transitivity.4 Informally, symmetry captures the idea that any two vertices can be
switched without the network changing its structure. To formalize the concept, we define an
automorphism on a graph to be a bijection on the vertices of the graph, f : N ↔ N , such that
(f (a), f (a )) ∈ E if and only if (a, a ) ∈ E. A graph is distance-transitive if for every quadruple
of vertices, a, a , b, and b , such that D(a, b) = D(a , b ), there is an automorphism, f , such
that f (a) = a and f (b) = b . An economy is said to be network symmetric if its graph is
distance-transitive. Preference symmetric economies and network symmetric economies provide
useful benchmarks to which the general class of economies can be compared.
We point out that network symmetry does not imply that the same amount of information
is diffused among agents at each point in time. It does, however, still impose severe restrictions
on how information may spread in the economy, as shown by the following lemmas:5
Lemma 1 In a network symmetric economy, ΔVa,t is the same for all agents at each point in
time. That is, for each t, for each a, ΔVa,t = ΔVt for some common ΔVt .
Thus, in a network symmetric economy, all agents have an equal precision of information at any
point in time, although their signal realizations of course differ.
Lemma 2 In a network symmetric economy the sequence ΔV1 , ΔV2 , . . . , ΔVT , is unimodal.
Specifically, there are times 1 ≤ t1 ≤ t2 ≤ T , such that ΔVt+1 > ΔVt for all t ≤ t1 , ΔVt+1 = ΔVt
for all t1 < t ≤ t2 , and ΔVt+1 < ΔVt for all t > t2 .
In other words, the typical behavior of the information diffusion process in a network symmetric economy is “hump-shaped,” initially increasing, after which it reaches a plateau and then
decreases.
3.3
Replica network
To justify the assumption that agents are price takers, the number of agents needs to be large.
Moreover, as analyzed in Ozsoylev and Walden (2011), restrictions on the distribution of number
Other notions include vertex transitivity, distance regularity, arc-transitivity, t-transitivity, and strong regularity, see Briggs (1993). Distance transitivity is a stronger concept than vertex transitivity, arc-transitivity, and
distance regularity, respectively, but neither stronger, nor weaker, than t-transitivity and strong regularity.
5
The first result follows immediately from the fact that automorphisms preserve distances between nodes, see
Briggs (1993), page 118. The second result follows from Taylor and Levingston (1978), where the result is shown
for the larger class of distance regular graphs (see also Brouwer, Cohen, and Neumaier 1989, page 167).
4
9
of connections are needed, to ensure existence of equilibrium. Ozsoylev and Walden (2011)
carry out a fairly general analysis of the restrictions needed on the degree distribution for the
existence of equilibrium to be guaranteed. They show that a sufficient condition is that the
degree distribution is not too fat-tailed. Compared with their static model, our model has
the additional property of being dynamic. Therefore, not only would restrictions on first-order
degrees be needed to ensure the existence of equilibrium, but also on degrees of all higher orders.
In the dynamic economy, signals spread over longer distances, thereby “fattening” the tail of
the distribution of signals among agents over time. We therefore believe that a general analysis
would be technically challenging, while adding limited additional economic insight, which is why
we choose the simplified approach.
We build on the concept of replica economies, originally introduced by Edgeworth (1881)
to study the game theoretic core of an economy (see also Debreu and Scarf 1964). We assume
that the full economy consists of a large number, M , of disjoint identical replicas of the network
previously introduced, and that agents’ random signals are independent across these replicas. A
replica network approach provides the economic and technical advantages of a large economy,
namely that price taking behavior is rationalized and that the law of large numbers makes most
idiosyncratic signals cancel out in aggregate, while avoiding the issues of signals spreading too
quickly among some agents, causing equilibrium to break down.
The total number of agents in the economy is N̄ = N × M . Formally, we define the set of
agents in an M -replica economy as Am = N × {1, . . . , M }, where a = (i, j) ∈ Am represents the
ith agent in the jth replica network, in an economy with M replica networks. There is still one
asset, one market, and one competitive market maker in the market with N̄ agents. We use the
enumeration a = 1, . . . , M N, of agents, where agent (i, j) maps to a = (j − 1)N + i.
Agent (i, j) and (i, j ) are thus ex ante identical in their network positions and in their
signal distributions, although their signal realizations (typically) differ. We let M increase in
a sequence of replica economies, with the natural embedding A1 ⊂ A2 · · · ⊂ Am ⊂ · · · , and
take the limit A = limM →∞ AM , letting A define our large economy, in a similar manner as in
Hellwig (1980). The network G is thus a representative network in the large economy, A. Our
interpretation is that the network, G, represents a fairly localized structure, perhaps at the level
of a town or municipality in an economy, whereas A represents the whole economy.
At time t, the market maker observes the average order flow per agent in the network6
wt = ut +
3.4
N̄
1 Δxa,t .
N̄ a=1
(3)
Equilibrium
We restrict our attention to linear equilibria in which agents in the same position in different
replica networks are (distributionally) identical. Such equilibria are thus characterized by the
Technically, the market maker observes ut + limM →∞
this can be done without confusion.
6
10
1
MN
M N
a=1
Δxa,t . We avoid such limit notation when
behavior of agents a = 1, . . . , N , who are “representative.” Our main existence result is the
following theorem, that shows existence of a linear equilibrium in the large economy under
general conditions and, furthermore, characterizes this equilibrium.
Theorem 1 Consider an economy characterized by M. For t = 1, . . . , T , define
At =
N
τ Va,t
,
N
γa
a=1
yt = τu (At − At−1 )2 ,
t
Yt =
ys ,
s=1
Ca,t =
Da,t =
τv + τ Va,t+1 + Yt+1
τv + τ Va,t + Yt
T
τv + Yt
τv + Yt+1
1 + τ Va,t
1
1
−
τv + Yt τv + Yt+1
,
−1/2
Ca,s
,
s=t+1
with the convention that A0 = 0, and YT +1 = ∞. There is a linear equilibrium, in which prices
at time t are given by
pt =
t
τv
Yt
τu v̄ +
v+
(As − As−1 )us .
τv + Yt
τv + Yt
τv + Yt s=1
(4)
In equilibrium, agent a’s time-t demand and expected utility, given wealth Wa,t and the realization
of signals summarized by za,t , take the form
xa,t =
τ Va,t
(za,t − pt ),
γa
−γa Wa,t − 12 τ
Ua,t = −Da,t e
(5)
2
τ 2 Va,t
(z −pt )2
+Y
+τ
Va,t a,t
v
t
.
(6)
Several observations are in place. First, note that the price function (4) has a fairly standard
structure. It is determined by the fundamental value (v) and the aggregate supply shocks (us ,
s = 1, . . . , t). The weights on these different components are determined by how signals spread
through the network. Especially, At summarizes how aggressive—and thereby informative—the
trades of agents are at time t, consisting of a weighted average of t-degree connectivity of all
agents. The variable Yt corresponds to a cumulative average of squared innovations in A up until
time t, and determines how much of the fundamental value that is revealed in the price. The main
generalization compared with Vives (1995) is that Va,t varies with agent and over time, depending
on the network structure. Moreover, preferences are allowed to vary across agents, through γa .
This allows us to compare the equilibrium dynamics that may arise because of heterogeneous
11
preferences with the dynamics that may arise because of heterogeneous information diffusion.
It is notable that Yt does not only depend on the total amount of information that has been
diffused at time t, but also on how this information has diffused over time. In other words
the price at a specific point in time is information path dependent. For example, consider two
economies with 4 agents, all with unit risk aversion (γa = 1), and with parameters τ = τv =
τu = 1. The first network, shown in panel A of Figure 1, is tight-knit (it is even complete) with
every agent being directly connected to every other agent. It is straightforward to calculate
1
. The second
V1,a = V2,a = 4, A1 = A2 = 4, Y1 = Y2 = 16, via (4) leading to p2 − v ∼ N 0, 17
network, shown in Panel B of Figure 1, is not as tightly knit, and agents have to wait until t = 2
before they have received all signals. In the latter case, V1,a = 3, V2,a = 4, A1 = 3, A2 = 4,
1
Y1 = 9, Y2 = 16, via (4) leading to p2 − v ∼ N 0, 11
. Thus, the price at t = 2 is less revealing in
the second case, even though all agents have the same information at t = 2 in both economies.
The reason is that in the tight-knit economy, the information revelation is more lumpy, whereas
it is more gradual in the less tight-knit economy. Lumpy information diffusion leads to more
revealing prices, since it generates more aggressive trading behavior in some periods, in turn
making it easier to separate informed trading from supply shocks. This is our first example of
how the network structure impacts asset price dynamics.
A.
B.
Figure 1: Impact of network structure. The figure shows two networks with four agents: In Panel
A, a tight-knit network is shown, in which every agent is connected with every other agent. In panel B, a
less tight-knit network is shown. At t = 2, prices are more revealing in the tight-knit network, since the
aggregate information arrival has been more lumpy, in turn leading to more revealing trading behavior
of informed agents.
4
Trading, profits, and centrality of individual agents
We study how the trades and performance of agents are determined by their positions in the
network.
12
4.1
Correlation of trades
Feng and Seasholes (2004) studied retail investors in the People’s Republic of China, and found
that the geographical position of investors was related to the correlation of their trades: geographically close investors had more positively correlated trades than investors who were farther
apart. Colla and Mele (2010) showed that information networks can give rise to such patterns
of trade correlations, under the assumption that geographically close agents are also close in the
information network. Their analysis was restricted to a cyclical network, but the effect was also
shown to arise in general networks in the static model of Ozsoylev and Walden (2011).
In Ozsoylev, Walden, Yavuz, and Bildik (2013), this positive relationship between trades and
network position was used to reverse engineer a proxy of the information network in the Istanbul
Stock Exchange from individual investor trades. Loosely speaking, agents who repeatedly traded
in the same stock, in the same direction, at similar points in time, were assumed to be linked in
the market’s information network. Such an approach is justified in the static model of Ozsoylev
and Walden (2011), but the situation is more complex in a dynamic setting. Specifically, one
may expect a positive relationship between network proximity and portfolio holdings also in
the dynamic model, since agents who are close in the network have many overlapping signals
and thereby similar information, leading to similar portfolio holdings. However, whereas agents’
trades and portfolio positions are equivalent in the static model, they are not in a dynamic
model. Instead, portfolio holdings are equivalent to cumulative period-by-period trades in the
dynamics model. For period-by-period trading behavior, the timing of information arrival is
also important, and this timing is different even for agents who are close in the network.
Consider, for example, an economy in which one agent, a, is connected to many other agents
at a distance t, and therefore receives very precise information about the asset’s value at time
t. This agent will at t take a large position in the asset (positive or negative, depending on
whether the agent believes that it is under- or over-valued). Moreover, assume that one of
agent a’s neighbors, agent b, does not have many connections within a distance of t but, being
connected to a, receives a lot of information at t + 1. Now, assume that at t + 1 there is also
a substantial amount of information incorporated into the asset’s price (driven by many other
agents who are also connected to many agents at distance t + 1). In this case, agent b will take
on a similar position as agent a at time t + 1, although less extreme since the price will be closer
to the fundamental value at this point. Agent a, however, may actually decrease his position, to
realize profits and decrease risk exposure. This argument suggests that the time t + 1 trades of
the two agents are negatively correlated, although they are neighbors in the network. Thus, the
period-by-period relationship between network proximity and correlation of trades seems less
clear than that between network proximity and portfolio holdings, suggesting that one needs
to be careful in choosing an appropriate window length when inferring network structure from
observed trades.
The following theorem characterizes covariances of trades, for an individual agent over time,
and between agents at a specific point in time.
13
Theorem 2 The covariance between an agent’s trade at t and t + 1 is
τ
Va,t
Va,t+1
−
.
Cov(Δxa,t+1 , Δxa,t ) = ΔVa,t
γa
τv + Yt+1 τv + Yt
(7)
The covariance of agent a and b’s trades at time t is
τ2
Cov(Δxa,t , Δxb,t ) =
γa γb
ΔVa,t ΔVb,t
yt
+ ΔVa,b,t .
Va,t−1 Vb,t−1 +
(τv + Yt−1 )(τv + Yt )
τv + Yt
(8)
Equation (7) shows that in the special case when an agent does not receive any new signals (and
thus ΔVa,t = 0), the covariance between time t and t + 1 trades is also zero. This is natural
since the agent’s trades in this case depends solely on price changes, and the price process is
a martingale. In the more interesting case when the agent does receive signals, the covariance
between subsequent trades is determined by the information advantage of the agent over the
market at t and t + 1, respectively. Specifically, Va,t represents how much information agent a
has received at time t, whereas τv + Yt represents how much aggregate information has been
incorporated into prices. A high Va,t relative to τv + Yt , means that the agent has a substantial
information advantage at time t. If the information advantage increases between t and t + 1,
then agent a’s trades will be positively correlated over these two periods, representing a situation
where he tends to ramp up investments. If, on the other hand, agent a’s information advantage
decreases, his trades will be negatively correlated, representing a situation where he takes home
profits and decreases risk exposure by selling stocks (or buying, if in a short position), in line
with our previous discussion.
An example of the two different situations is shown in Panels A and B of Figure 2. In Panel
A, agent a is at the center of an extended star network and will have a large advantage over
the market at t = 1. At t = 2, the playing field is more even, since information diffusion has
made all of agent a’s neighbors well-informed too. Agent a still has an information advantage,
now having received all signals from the periphery of the network, but the advantage is lower
than in the previous period, and he therefore decreases his asset position, leading to a negative
correlation of his trades over the two periods. In Panel B of the figure, agent a still receives
the same signals period-by-period, but since his neighbors are now more directly connected, his
information advantage at t = 1 is smaller. In this case, his information advantage may be higher
in period 2 than in period 1, so he may tend to gradually ramp up up his position over the two
time periods and therefore have positively correlated trades.
We note that this intuition, that relative information advantage over time determines an
agent’s dynamic trading behavior, which is very clear in the general setting, does not come
out clearly if we restrict our attention to symmetric networks. Indeed in a network symmetric
economy, ΔVa,t is the same for all agents, and it can be shown that (7) takes the form
τ
τ
Cov(Δxa,t+1 , Δxa,t ) = ΔVt+1 ΔVt τv + Yt − Vt ΔVt+1 .
γa
γ̄
14
Thus, all else equal, for low degrees of information diffusion between t and t + 1 (i.e., a low
ΔVt+1 ), trades will be positively correlated, whereas they will be negatively correlated when the
degree of information diffusion is high (i.e., for a high ΔVt+1 ). This result is the opposite of
what we just showed.
The issue in the symmetric case is that within the class of network symmetric economies,
connectivity must increase for all agents at the same time, so there is no way to increase the
relative information advantage of one agent. Increasing the connectivity of all agents at the same
time has two effects: it increases the total informativeness of their signals and it increases the
amount of information that is incorporated into the asset’s price, and the second effect always
dominates. Thus, increased connectivity at t + 1, all else equal, always decreases the information
advantage of the agents in a symmetric network setting, making them decrease their portfolio
holdings, and thereby potentially leading to negative correlation with their trades in the previous
period. This is our first example of how restricting ones’ attention to symmetric economies may
be misleading—in this case reversing the result.
We next focus on Equation (8), which shows how the time-t trades of two agents are related.
The first two terms in the expression represent covariance induced by the fact that two informed
agents will tend to trade in the same direction because, both being informed, they will take
a similar stand on whether the asset is over-priced or under-priced. This part of the expression increases in the total amount of information the agents have received at t − 1 (through
Va,t−1 Vb,t−1 ), as well as in how much additional information they expect to receive between
t − 1 and t (through ΔVa,t ΔVb,t ). Offsetting these effects is the aggregate informativeness of the
1
, similarly to what we saw in (7). The third
market, through the terms (τv +Yt−1yt)(τv +Yt ) and τv +Y
t
term in the expression provides an additional positive boost to the covariance, and is increasing
in the number of common agents at distance t of both agent a and b. This term is zero if the
agents are further apart than a distance of 2t, but will otherwise typically be positive. The term
captures the natural intuition that agents who receive identical information signals have more
similar trades than agents who receive signals with independent error terms.
The first main implication of (8) is that the covariance is always strictly positive. Thus, the
situation with negative correlation between trades—because two nearby agents tend to trade
in different directions when one is ramping down portfolio exposure whereas the other one is
ramping up—never occurs in the model. To understand why this is the case, we use (5) to
rewrite agent a’s time-t demand as
τ
j∈ΔSa,t sj
− pt − Va,t (pt − pt−1 ) .
ΔVa,t
Δxa,t =
γa
ΔVa,t
The first term in this expression represents the agent’s demand because of additional information
received between t − 1 and t. We note that j∈ΔSa,t sj /ΔVa,t = v + ζ a , where the error term
ζ a ∼ N (0, σ 2 /ΔVa,t ) is independent of prices. The second term represents the agent’s sloping
demand curve, causing him to rebalance portfolio holdings when the price catches up, by selling
(buying) stocks when the price increases (decreases) between t − 1 and t. For an agent who has
15
an information advantage at time t − 1, but receives no new information between t − 1 and t,
this second term is the only one present (since ΔVa,t = 0).
Now, agent b’s demand function has the same form as agent a’s and, assuming that agent
b receives a lot of new information between t − 1 and t, the first term dominates. Negative
correlation would then arise if agent b tends to ramp up when agent a ramps down, which
is the case if Cov v + ζ b − pt , −(pt − pt−1 ) < 0. However, since the market is semi-strong
form efficient, v − pt is independent of pt − pt−1 . Furthermore, since ζ b is independent of
aggregate variables, Cov ζ b , −(pt − pt−1 ) = 0. In other words, since agent a’s rebalancing
demand between t − 1 and t is publicly known at t, it must be independent of agent b’s time-t
demand which in turn is due to informational advantage at time t.
This result is model dependent. It depends on the linear structure of agents’ demand functions. However, to a first order approximation we expect the result to hold in more general settings in semi-strong form efficient markets, given that trading for rebalancing purposes mainly
depends on price changes, trading for informational purposes depends on the difference between
the true value and market price, and the two terms are uncorrelated in a weak-form efficient
market.
The second main implication of (8) is that, all else equal, period-by-period covariances (and
correlations) between two agents’ trades increase in the number of common acquaintances they
have (through the ΔVa,a ,t -term). As an example, in Panel C of Figure 2, we show a cyclical
network. It immediately follows that the correlation between two agents’ trades at any time
t < 4 (at which point all information has reached all agents) is decreasing in the distance between
the two agents.
These properties of trade correlations suggest that it may be justified to use trades instead
of portfolio holdings to draw inferences about a market’s information network as, e.g., done in
Ozsoylev, Walden, Yavuz, and Bildik (2013), using short time horizons over which trades are
compared.
A.
C.
B.
a
a
Figure 2: Correlation of trades. The figure shows three networks that lead to different types of
trading behavior. Panel A shows a network in which agent a’s trades at time 1 and 2 are negatively
correlated, whereas they are positively correlated in Panel B. Panel C shows a network in which the
correlation between two agents’ trades, at each point in time, is inversely related to the distance between
the two agents.
16
4.2
Profits
Who makes profits in an information network? Our starting point is the following theorem:
Theorem 3 Define
πa,t = τ Va,t
πa,T
= τ
1
1
−
τv + Yt τv + Yt+1
Va,T
,
τv + YT
,
t = 1, . . . , T − 1, a = 1, . . . , N
a = 1, . . . , N.
The ex ante certainty equivalent of agent a is
Ua =
1
log(C),
2γa
where C =
T
(1 + πa,t ) .
(9)
t=1
The expected profit of agent a is
τ
Πa ,
γa
where
Πa =
T
(τv + Yt )−1 Va,t
(10)
(11)
t=1
is the profitability of agent a.
We focus on profitability, and leave welfare implications of the model for future research.
Equation (10) determines (ex ante) expected profits of an agent. It shows that expected profits
depends on three components. First, profits are inversely proportional to an agent’s risk-aversion,
γa , because more risk-averse agents take on smaller positions—all else equal. This follows
immediately, since an agent’s equilibrium trading position is proportional to γa , so it corresponds
to pure scaling. Therefore, we do not include it in our measure of profitability, as defined by
Equation (11). Neither do we include the signal precision, τ , which is constant across agents.
Second, expected profits depend on an agent’s position in the network through {Va,t }t , t ∈ T : the
higher any given Va,t is, the higher the agent’s expected profits. Third, expected profits depend
inversely on the amount of aggregate information available in the market, in that at any given
point in time, the higher the total amount of aggregate information, the lower the expected
profits of any given agent. The third part represents a negative externality of information.
Equation (11) thus provides a direct relationship between the properties of a network, local as
well as aggregate, and individual agents’ profitability.
17
4.3
Centrality
Equation (11) shows that an agent’s profitability is determined by his centrality, defined appropriately. Recall that Va,t denotes the number of agents that are within distance t from agent a.
So, Va,1 is simply the degree of agent a. For t > 1, higher order connections are also important
in determining profitability. For example, Va,2 does not only depend on how connected agent a
is, but also on how connected his neighbors are. We use (1) to rewrite (11) on vector form as
Π=
∞
βt χ(E t )1,
(12)
t=1
where βt = (τv + Yt )−1 , for t ∈ T , and βt = 0 for t > T . Here, Π is an N -vector where the ath
element is the profitability of agent a.
We explore the relationship between this profitability measure and standard centrality measures in the literature. Specifically, we define degree centrality, farness, closeness centrality, Katz
centrality, and eigenvector centrality (see, e.g., Friedkin (1991) for a detailed discussion of these
concepts), and compare (12) with these measures.
Definition 3
• The degree centrality vector is the vector of first-order degrees, V 1 , i.e., the degree centrality of agent a is Va,1 .
• The farness of agent a is the agent’s average distance to all other agents, i.e.,
a =a D(a, a )
.
Fa =
N −1
• The closeness centrality of agent a, Ĉa , is the inverse of that agent’s farness,
Ĉa =
1
.
Fa
• The Katz centrality vector with parameter α < 1 is the vector K ∈ RN
+ , defined as
K = Kα =
∞
αt E t 1.
(13)
t=1
Here, E t denotes the t:th power of the adjacency matrix, E, and 1 ∈ RN is an N -vector
of ones.
• The eigenvector centrality vector is the eigenvector corresponding to the largest eigenvalue of E, i.e., the vector C that solves the equation C = λEC, for the largest possible
eigenvalue, λ, where we normalize C such that a∈N Ca = 1.
18
We see that the structures of the profitability measure (12) and Katz centrality (13) are similar.
Specifically they are both made up by a weighted sum of powers of the adjacency matrix,
multiplied with the vector of ones. The differences are that the weighting is a power of α for
Katz centrality but varies more generally with t for profitability, and that the matrix indicator
function, χ, operates on the power of the adjacency matrix in the profitability measure. It is
a standard result that eigenvector centrality can be viewed as a special case of Katz centrality,
α
since C = limαλ−1 KK α .7
a
a
The similar structure of the formulas for profitability (12) and Katz centrality (13) suggests
that Katz centrality is closely related to profitability, as is eigenvector centrality being a special
case of Katz centrality. Therefore, we may expect Katz and eigenvector centrality to dominate
the other measures for “an average” network, although there may be exceptions.
To formalize this intuition, we introduce a random network generating process, which allows
us to derive a probabilistic ranking of the different measures. We use the classical concept
of random graphs, originally studied in Erdös (1947), Erdös and Rényi (1959), and Gilbert
(1959), in which links between agents are formed randomly, independently, and with constant
probability.8
Our focus is on sparse networks—in line with what is observed in practice (see the discussion
in Ozsoylev, Walden, Yavuz, and Bildik (2013)). We assume that the expected number of links
per agent in a network of size N is c log(N )k + 1 for some c > 0 and k > 3.9 With this
assumption, the connectedness of agents grows with N , but the fraction of expected connections
in the network (which is approximately cN log(N )k /2) to maximum number of connections
(which is approximately N 2 /2) tends to zero for large N , so when N is large the network is
indeed sparse. To this end, we define
Definition 4 The Erdös-Rényi random graph model of size N , and parameters c > 0, and
k > 3, G(N, c, k), is defined so that for each 1 ≤ a ≤ N , 1 ≤ a ≤ N , a = a , (a, a ) ∈ E with
)k
i.i.d. probability c log(N
N −1 .
We use cross sectional correlation across agents as the metric of similarity.10 Specifically,
for a given network we define the cross sectional correlation between the different centrality
measures and profitability,
ρD = Corr(D, Π), ρF = Corr(−F, Π), ρĈ = Corr(Ĉ, Π), ρK α = Corr(K α, Π), ρC = Corr(C, Π).
7
Uniqueness of the eigenvector centrality measure is not guaranteed, but is almost never an issue in practice.
We focus on the Erdös-Rényi model, since it is the most parsimonious and analytically most tractable. Many
other network generating processes have also been suggested in the literature, e.g., the preferential attachment
model of Barabasi and Albert (1999), and the model of Watts and Strogatz (1998).
9
The restriction k > 3 is needed for technical reasons.
10
2
2
The cross sectional statistics are defined as EX = N1
a∈N Xa , σ (X) = E[(X − EX) ], Cov(X, Y ) =
E[(X − EX)(Y − EY )], and Corr(X, Y ) = Cov(X, Y )/(σ(X)σ(Y )).
8
19
Here, we use the convention that the correlation between any random variable and a constant is
zero. In the random network model, these cross sectional correlations are in turn random, since
they depend on the random realization of the network. Note that ρF is defined as the correlation
between negative farness and profits, since farness is inversely related to connectedness.
For simplicity, we focus on preference symmetric economies, but allow for γ, τ , τu , τv , v̄, and
T to be arbitrary. The following result shows that Katz centrality dominates degree centrality,
farness, and closeness centrality for large random graphs.
Theorem 4 For any given c > 0 and k > 3, there is an N0 , such that for all networks of size
N > N0 in the G(N, c, k) model, there is an α, such that
E [ρK α ] > E [ρD ] > max E ρĈ , E [ρF ] .
(14)
Thus, for large random networks, profitability is best characterized by Katz centrality. This result provides a strong ranking between different centrality measures for a large class of networks.
To the best of our knowledge, this is the first such ranking of centrality measures in equilibrium
models of asset pricing with general networks. In contrast, previous applications of network
theory to asset pricing typically provide results for very specific networks (e.g., Ozsoylev 2005,
Colla and Mele 2010, and Buraschi and Porchia 2012), or is empirically motivated (e.g., Das and
Sisk 2005, Adamic, Brunetti, Harris, and Kirilenko 2010, Li and Schurhoff 2012, and Ozsoylev,
Walden, Yavuz, and Bildik 2013,). The challenge in proving the result stems from the fact that
profitability is determined endogenously in equilibrium.
The intuition behind the result is straightforward, as shown in the proof of the theorem,
suggesting that it may extend to more general settings. The profitability equation (12) is
made up by a weighted average of the connectedness of different orders (with the weights,
βt , endogenously determined). Degree centrality exclusively focuses on first-order connections,
whereas closeness centrality and farness mainly focus on high-order connections (since the vast
majority of agents will be quite far away from any given agent in a large network). Katz and
eigenvector centrality, in contrast, balance the weights of different orders of connectedness, and
are thereby more closely related to profitability.
Of course, our strong ranking of centrality measures holds only for large networks. We also
explore the relationship between the different centrality measures in medium-sized networks
using simulations, to verify that Katz centrality also works well in such networks. We randomly
simulate 1,000 economies of sizes N = 50, 100, 200, 500, and 1000, respectively, and in each
economy we randomly generate links between agents, using the random graph model, so that
√
each agent is expected to have N links. The growth of the number of links in the network with
N is thus slightly faster than in the G(N, c, k) model, although the graph is still asymptotically
sparse.
We measure the correlation between profitability (Π) and the different centrality measures.
The results are shown in Table 1. We see that the ranking is the same as in Theorem 4 and,
20
Size of network, N
50
100
200
500
1000
Mean correlation
C
0.96
0.94
0.94
0.95
0.97
0.95
Kα
0.96
0.94
0.94
0.95
0.97
0.95
V1
0.88
0.82
0.79
0.83
0.85
0.83
Ĉ
0.0
0.0
0.0
0.0
0.0
0.0
F
0.0
0.0
0.0
0.0
0.0
0.0
Table 1: Centrality Measures. The table shows the correlation between profitability, Π, and centrality
for several different centrality measures, degree centrality (V 1 )), eigenvector centrality (C), Katz centrality (K α ), closeness centrality (Ĉ) and farness (F ). The parameters of the economy are σ = σu = σv = 10,
v̄ = 0, T = 5, and γ = 1 for all agents. The number of agents is varied between N =
√ 50 and N = 1000,
and links are randomly drawn between agents, such that on average an agent has N links. For each
N , we simulate 1, 000 random networks. The results show that eigenvector and Katz centrality are most
closely related to profitability, followed by degree and betweenness centrality. Closeness centrality and
farness perform poorly, since there is usually an isolated agent in the network, leading to all agents having
closeness centrality of zero and infinite farness centrality.
furthermore, that eigenvector centrality, being a special case of Katz centrality, performs about
as well as the Katz centrality measure.
5
Aggregate volatility and trading volume
According to the Mixture of Distributions Hypothesis (MDH), return volatility varies over time,
which then leads to heavy-tailed unconditional return distributions. A common explanation for
such time varying volatility—as well as trading volume—is lumpy diffusion of information into
the market (Clark 1973; Epps and Epps 1976; Andersen 1996).
Rich dynamics of volatility and trading volume have indeed been documented in the literature. First, trading volume is positively autocorrelated over extended periods, i.e., the autocorrelation function of trading volume has “long memory” in that it decreases very slowly
(see Bollerslev and Jubinski 1999 and Lobato and Velasco 2000). Roughly speaking, this means
that abnormally high trading volume one period predicts abnormally high trading volume for
many future periods. Second, return volatility of individual stocks, and markets, also have long
memory (e.g., Bollerslev and Jubinski 1999 and Lobato and Velasco 2000). Third, volume and
volatility are related (Karpoff 1987; Crouch 1975, Rogalski 1978), in line with the Wall Street
wisdom that it takes volume to move markets. Contemporaneously, trading volume and absolute price change are highly positively correlated. The two series also have positive lagged
cross-correlations. Using a semi-parametric approach to study returns and trading volume on
the NYSE, Gallant, Rossi, and Tauchen (1992) show that large price movements predict large
trading volume. In other markets, there is evidence for a reverse casualty, i.e., that large trading volume leads large price movements. For example, Saatcioglu and Starks (1998) find such
evidence in several Latin American equity markets.
21
In our model, agents’ preferences (Γ) and the network structure (E) determine volatility
and volume dynamics in the market. It is then natural to ask which type of dynamics can be
generated within the model for general Γ and E, and also what one can infer about the network
structure from observed volatility and volume dynamics.
5.1
Volatility
In a general network economy, we would expect price volatility to vary substantially over time.
For example, information diffusion may initially be quite limited, with low price volatility as
an effect, but eventually reach a hub in the network, at which point substantial information
revelation occurs with associated high price volatility. The following result characterizes the
price volatility over time, and moreover shows that any volatility structure can be supported in
a general economy.
Theorem 5 For t = 1, · · · , T , the variance of prices between t − 1 and t, is
2
=
σp,t
yt
,
(τv + Yt )(τv + Yt−1 )
(15)
where we use the convention that Y0 = 0, and between T and T + 1, it is
2
σp,T
+1 =
1
.
τv + YT
(16)
+1
kt = 1 and an arbiMoreover, given coefficients, k1 , . . . , kT +1 , such that kt > 0, and Tt=1
trarily small > 0, there is a preference symmetric economy, such that
2
σp,t − kt ≤ ,
t = 1, . . . , T + 1.
(17)
τv From the first part of the theorem, we see that the volatility (the square root of variance) has
a general decreasing trend over time because of the increasing denominator in (15), but that it
can still have spikes in some time periods because of large values in the numerator. In fact, (17)
shows that any structure of time-varying volatility after an information shock can be generated.
Note that the total cumulative variance up until time t ≤ T is
2
=
σP,t
1 Yt
.
τv τv + Yt
(18)
This part of the variance that is incorporated until T represents the “information diffusion”
component of asset dynamics, whereas the part between T and T + 1, i.e., (16), represents the
component due to public information sources, in line with the discussion in Section 3.2. The
22
total price variance between 0 and T + 1 is of course equal to the ex ante variance,
2
2
+ σp,T
σv2 = σP,T
+1 =
1 YT
1
+
,
τv τv + YT
τv + YT
independently of network structure. But the way the variance is divided period-by-period, and
into the information diffusion and public component, depends on the network. It is specifically
determined by yt , t = 1, . . . , T .
If we restrict our attention to network symmetric economies, the possible dynamic behavior
of volatility is much more restricted. This is not surprising, given the restrictions on information
diffusion dynamics described in Lemmas 1 and 2. From (18), it follows that yt is proportional
1
2
to Δηt = ηt − ηt−1 , where ηt = σ2 −σ
2 . Since yt is proportional to ΔVt (which is the same for
v
P,t
all agents in a network symmetric economy), ΔVt is unimodal, and the square of a nonnegative
unimodal function is also unimodal, it follows that the sequence Δη is also unimodal. Moreover,
2
if the public information component is large compared with the diffusion component, σP,T
<<
2
σv , it follows from (18) that
2
,
(19)
σv4 Δηt ≈ σp,t
so in this case σp,t is also unimodal. We summarize this in
Corollary 1 In a network symmetric economy,
1. Δηt is unimodal,
2
<< σv2 , then σp,t is unimodal.
2. if the public information component is high, σP,T
Of course, the timing of an information shock is typically not known. We have in mind
a situation in which information shocks arrive every now and again, and are then gradually
incorporated into prices. We do not observe the timing of these shocks, and therefore do not
know when t = 0 in our model.
However, autocorrelations — which take averages over all time periods — can be calculated
even without observing the timing of shocks. Now, there is a close relationship between a function’s shape and the shape of its autocorrelation function. For a general sequence, f0 , f1 , . . . , fT ,
we define the difference operator (Δf )k = fk − fk−1 , k = 1, . . . , T , (Δf )T +1 = 0 − fT , and the
higher-order operators Δn+1 f = Δ(Δn f ). The following lemma relates the monotonicity of a
function and its autocorrelation function:
Lemma 3 Consider a nonnegative sequence f0 , f1 , . . . , fT , such that f0 = 0. Define the auto −k
fk fk+|τ |, −T + 1 ≤ τ ≤ T − 1. Then,
correlation function Rτ = Tk=0
1. if f is unimodal, so is R,
2. if Δn f does not switch sign, then R has at most n − 1 turning points.
23
In light of Corollary 1, the result immediately leads to:
Corollary 2 In a network symmetric economy with a high public information component, the
autocorrelation function of volatility is unimodal.
Thus, given the restrictions imposed by network symmetry, nonmonotonicity of the autocorrelation function of volatility suggests that there is asymmetry in the underlying information
network.
One way of measuring the degree of monotonicity of an autocorrelation function is by counting its number of turning points for positive τ (since the autocorrelation function is symmetric, it
is sufficient to study τ > 0). Another way is to measure its absolute variation, τ >0 |Rτ +1 −Rτ |,
which will be higher the more nonmonotone the autocorrelation function.
To summarize, network symmetry leads to hump-shaped volatility after an information
shock, and also to hump-shaped autocorrelation functions, whereas any dynamics can arise
in a preference symmetric economy.
5.2
Volume
Just like with volatility, rich dynamics of trading volume can arise within the network model.
Since the model is inherently asymmetric, aggregate trading volume will be made by the heterogeneous trades of many different agents. This contrasts to the uniform behavior in models with
a representative informed agent (e.g., Kyle 1985), as well as to the ex ante symmetric behavior
in economies with symmetric information structures (e.g., Vives 1995; He and Wang 1995).
We focus on the aggregate period-by-period trading volume of agents in the network, since
the stochastic supply is (quite trivially) normally distributed. To this end, we define:
Definition 5 The time-t aggregate (realized) trading volume is Wt =
ante) expected trading volume is Xt = E [Wt ].
1
N
a |Δxa,t |,
and the (ex
The following theorem characterizes the expected trading volume, and mirrors our volatility
results by showing that any pattern of expected trading volume can be supported in the model.
Theorem 6 The time-t expected trading volume is
Xt =
N
2
2
2
V
V
ΔV
ΔV
τ
1
2
a,t−1
a,t
a,t
a,t
.
−
+
+
N a=1 γa π τv + Yt−1 τv + Yt τv + Yt
τ
24
(20)
Given positive coefficients, c1 , c2 , . . . , cT +1 , and any > 0, there is an economy such that
|Xt − ct | ≤ ,
t = 1, . . . , T + 1.
It is clear from (20) that, as is the case for volatility, heterogeneous preferences alone cannot
generate such complete generality of trading volume dynamics. In a network symmetric economy,
all terms under the square root are identical across agents, and (20) collapses to
Xt =
2τ 2
πγ̄ 2
2
Vt−1
Vt2
ΔVt2
ΔVt
−
+
+
.
τv + Yt−1 τv + Yt τv + Yt
τ
(21)
Again, differences in preferences in this case are only important through the effect they have on
the (harmonic) average risk aversion coefficient, γ̄. Moreover, the restrictions on ΔVt imposed by
network symmetry carry over to trading volume. It is easily seen that if the public information
component is large compared with
the diffusion component, the fourth term under the square
2τ 2
root in (21) dominates and Xt ≈ πγ̄
2 ΔVt . In this case, we therefore get:
Corollary 3 In a network symmetric economy, in which the public information component is
high:
• Xt is unimodal, as is its autocorrelation function,
• There is an approximate square root relationship between σp,t and Xt :
Xt ∼
√
σp,t .
(22)
To summarize, the degree of nonmonotonicity of the autocorrelation functions of volatility
and trading volume are related to the degree of asymmetry in the information diffusion process.
In the network symmetric case, after an information shock trading volume and price volatility
increase, reach a peak and then decrease. In the asymmetric case, the dynamics of volume and
volatility after an information shock may be nonmonotone. Also, the instantaneous correlation
of volatility and volume is high in the symmetric case.
In the completely general case, with heterogeneity over both preferences (γa ) and network
structure (Va,t ), we expect the interplay between the two to give rise to quite arbitrary trading
volume and volatility dynamics. For example, at a large time, t, almost all information may
have diffused among the bulk of agents, leading to a small and decreasing ΔVa,t , and thereby low
volatility. A peripheral agent with very low risk aversion, who receives many signals very late,
may still generate large trading volume at such a late point in time, despite the low volatility.
The above example captures the important distinction between trading volume driven by
high aggregate information diffusion, and by demand from agents with low risk aversion, a
25
distinction that does not arise in either the preference symmetric or the network symmetric
benchmark cases.
5.3
Conditioning on event time
In the previous analysis we studied unconditional relations, assuming that the arrival time of
information shocks is unknown. In some cases the arrival time may be observable. Earnings
announcements may, for example, constitute well defined information events for which we could
use such conditioning. Although the actual announcement is a public event, if agents have
additional private information, the inferences they draw from the announcement together with
their private information may differ.
If the time of arrival of the information shock is well-identified, we can condition on the
time that has passed since arrival and calculate conditional relations between returns and volume. Specifically, we can use t in our information set when calculating relations, e.g., defining
Cov(Wt , Wt+1 ) = E[(Wt −E[Wt ])(Wt+1 −E[Wt+1 )]|t]. We define the price change, μt = pt −pt−1 ,
and we then have
Theorem 7 Trading volume and price changes satisfy the following conditional relations:
• Corr(Wt , |μt |) > 0,
• Corr(Wt , Wt−1 ) > 0,
• Wt is independent of μs for s < t.
The first two results state that trading volume is contemporaneously positively related to absolute price changes, as well as positively autocorrelated. These results are in line with the
previously discussed empirical literature. The third result states that price movements do not
cause lagged trading volume. The result seems to be inconsistent with what is found in Gallant,
Rossi, and Tauchen (1992), but consistent with the results in Saatcioglu and Starks (1998). We
emphasize though that to test this prediction, one needs to condition on the time elapsed after
an information event, which these studies do not do. It is therefore an open question whether
the prediction holds empirically.
5.4
Size versus volatility and volume
The results in Sections 5.1 and 5.2 can potentially be used to understand variations in dynamics
across different markets and assets. One such variation is between companies of different size,
where we may expect more transparent information diffusion for large than for small companies, because of the higher coverage of large companies. The underlying information network
structure for large companies may therefore be relatively symmetric, and the public information
26
component may be large. In contrast, the network structure of smaller companies may be more
asymmetric, and the nonpublic information component more significant.
We carry out initial tests of whether the observed dynamics of volatility and volume in the
market is consistent with such differences between large and small companies. The tests are
suggestive and exploratory. A more extensive empirical investigation, e.g., in the form of reverse
engineering the underlying network using maximum likelihood estimation, is outside of the scope
of this paper.
5.4.1
Data
Using the Center for Research in Security Prices (CRSP), we collect daily returns, market value,
and trading volume over a ten-year period (January 2003-December 2012), for all companies in
the Russell 3000 Index. We classify the companies as Large Cap (market capitalization above
USD 10 Billion), Mid Cap (market capitalization between USD 1 Billion and 10 Billion), and
Small Cap (market capitalization below USD 1 Billion). We require data for at least 98% of
the days in the sample (2,466 of the 2,517 trading days) to be available, to include a company
in the sample, leaving us with 276 Large Cap, 746 Mid Cap, and 502 Small Cap companies, in
total J = 1524 companies.
We calculate weekly return volatility and turnover for each stock in the sample, which we
σ , and turnover, RW , for each stock
then use to calculate weekly autocorrelations of volatility, Rj,t
j,t
(j), and lag t = 1, 2, . . . , 12 weeks. We use turnover instead of raw volume, to get a normalized
trading volume measure.
5.4.2
Results
We calculate the average number of turning points of the autocorrelation functions for volatility
and turnover, in the three groups. The results are shown in Table 2. We see that the number of
turning points is larger for smaller firms, both for autocorrelation of volatility and of turnover.
The differences are statistically significant at the 2.5% level (t-statistic of 2.02, using a one-sided
test) when testing for differences of means between Mid Cap and Small Cap companies, whereas
it is significant at the 0.1% level for the other three tests (Large Cap verses Mid Cap, and Mid
Cap versus Small Cap for volatility, and Large Cap versus Mid Cap for turnover). This is in
line with our analysis, that large companies have more symmetric information networks with a
larger public component than small companies.
A similar picture emerges for absolute variation of the autocorrelation functions of volatility
and volume, for firm j defined as
12
σ
|Rj,t
−
σ
Rj,t−1
|
and
t=1
12
W
W
|Rj,t
− Rj,t−1
|,
t=1
respectively. As seen in Table 2, the average absolute variation is higher for smaller companies,
with similar statistical significance as for the number of turning points. Since absolute variation
27
Large Cap
σ
A. Volatility, Rj,t
Average number of turning points
Mid Cap
3.63
4.08
(3.76)
Average absolute variation
0.88
0.92
3.70
4.18
0.71
0.77
0.57
0.85
(6.70)
0.53
(-4.00)
Number of firms
4.37
(2.02)
(3.90)
C. Correlation
0.99
(5.67)
(3.74)
Average absolute variation
4.46
(3.81)
(2.03)
W
B. Turnover, Rj,t
Average number of turning points
Small Cap
276
0.41
(-14.1)
746
502
Table 2: Company size versus dynamics of volatility and turnover. Panel A shows how the
σ
, is related to company size. Companies are divided into three
autocorrelation function of volatility, Rj,t
groups: Large Cap (market capitalization above USD 10 Billion), Mid Cap (market capitalization between
USD 1 and 10 Billion), and Small Cap (market capitalization below USD 1 Billion). Average number of
turning points, and average absolute variation are shown in rows 1 and 3, respectively. The t-statistic of
a difference of means test between number of turning points of Large Cap and Mid Cap companies, and
between Mid Cap and Small Cap companies, are shown in columns 2 and 4 of row 2, respectively, and the
same difference in means tests for absolute variation in rows 2 and 4 of row 4. Panel B shows the same for
the autocorrelation function of turnover. Panel C shows the average correlation between volatility and
turnover for the three groups of companies. Columns 2 and 4 show the t-statistics of difference in means
test between Large and Mid Cap companies, and between Mid and Small Cap companies, respectively.
provides another measure of nonmonotonicity, this reinforces the view that smaller stocks are
associated with more asymmetric information networks than large stocks.
Finally, the cross correlation between volatility and turnover is higher for Large Cap companies than for Small Cap companies (0.57 for Large Cap versus 0.41 for Small Cap, with
differences in means strongly statistically significant both between Large Cap and Mid Cap, and
between Mid Cap and Small Cap). We argued earlier that in a symmetric network with a large
public information component, there will be a close instantaneous link between volatility and
volume, so this result is also consistent with less asymmetric networks for large stocks.
One potential issue with the test above is that trading volume is known to be non-stationary,
increasing over time (see, e.g., Lo and Wang 2000). Although we use turnover in our test, which
did not seem to have an identifiable trend during the sample period (the average weekly turnover
was 2.8% in the first year, 2003, peaked around 10% at the height of the financial crises, 2008,
and was then down to 3.4% in the last year of the period, 2012), we wish to control for trends
in our test, for the sake of robustness. We therefore carry out the same tests as in Table 2,
but normalize volatility and turnover by dividing these variables by the average volatility and
turnover across all stocks, in any given week. Thus, the sum of normalized volatilities and
turnovers in any given week is equal to one. The results (not reported) are virtually identical
to those in Table 2, so non-stationarity does not see to be driving our results.
Another implication of the model is that in network symmetric economies with large public
28
information components, expected trading volume scales as the square root of volatility (22).
In log-coordinates, we can write
wj,t = aj + bj sj,t + j,t ,
(23)
where wj,t = log(Wt ) is the logarithm of trading volume, and sj,t = log(σp,t ) is the logarithm of
volatility, for firm j at time t, j,t are random shocks, and the coefficient bj = 0.5 for firms with
symmetric networks and large public information components. We investigate this relation.
For each firm, we regress log-turnover on log-volatility, and we then calculate the average bj
coefficient for firms in the three different groups. The results are shown in Table 3, Panel A. We
see that the average coefficients are lower than 0.5, although quite close, for all groups. They
are about 0.43 for the Large Cap and Medium Cap groups, and 0.38 for the Small Cap group.
This suggests that the relationship between volatility and volume is farthest away from a square
root relationship for stocks in the Small Cap group, again consistent with more asymmetry in
this group. The difference in means between Mid Cap and Small Cap companies is significant,
with a t-stat of -5.67.
The regressions in (23) do not separate between aggregate shocks to volatility and volume, and company specific shocks. If shocks are mainly aggregate, a cross sectional comparison between companies will allow for only limited inferences. We can see how well volatility
andturnover are
regressing log-average turnover, w̄t =
related in the aggregate market, by 1 1
log J j Wt (j) , on log-average volatility, s̄t = log J j σp,t (j) , i.e.,
w̄t = ā + b̄s̄t + t .
The b̄ coefficient in this regression is insignificantly different from 0.5 (see Panel C of Table 3),
suggesting that the square root relationship holds well at the market level.
To focus on excess turnover and volatility, above and beyond market shocks, we regress
wj,t − w̄t = aej + bej (sj,t − s̄t ) + ej,t .
The average bej coefficients for companies in the three groups are shown in Panel B of Table 3. The
coefficients are lower for all groups compared with those in Panel A—between 0.32 and 0.35—as
are the R-squares, suggesting a significant market component that generates both volatility and
volume for individual stocks. The lowest average coefficient is again for companies in the Small
Cap group, significantly lower than the average of the Mid Cap companies (t-statistic of -2.51).
However, the average coefficient for the Large Cap companies is in this case lower than for
the Mid Cap companies. Altogether, the results are consistent with the following story: Large
companies have symmetric information networks, and the type of information that is relevant
for these companies is also closely related to aggregate market information, causing their excess
regression coefficient to be low. Mid-sized companies also have symmetric networks, but their
information is not as closely related to market information, i.e., it is more idiosyncratic. Finally,
29
A. Log-regression
Average b
Large Cap
Mid Cap
Small Cap
0.43
(0.009)
0.43
(0.005)
0.38
(0.008)
(0.35)
Average R2
B. Excess log-regression
Average be
(-5.67)
0.30
0.25
0.15
0.33
(0.011)
0.35
(0.0065)
0.32
(0.0096)
(1.64)
Average R2
C. Market regression
Market b̄
R2
D. Correlations
Average ρ(s̄, sj )
Average ρ(w̄,wj )
0.17
Market
(-2.51)
0.15
0.11
0.51
(0.022)
0.52
0.63
0.60
0.60
0.54
0.54
0.41
Table 3: Relationship between log-turnover and log-volatility. Panel A shows how the regression
coefficient of log-turnover on log-volatility is related to company size. Companies are divided into three
groups: Large Cap (market capitalization above USD 10 Billion), Mid Cap (market capitalization between
USD 1 and 10 Billion), and Small Cap (market capitalization below USD 1 Billion). Average regression
coefficients are shown in row 1, and standard deviations in row 2. The t-statistic of difference of means
tests between regression coefficients of Large Cap and Mid Cap companies, and between Mid Cap and
Small Cap companies, are shown in columns 2 and 4 of row 3, respectively. The average R-squares are
shown in Row 4. Panel B shows the same for excess log-turnover (defined as wj,t − w̄t ) regressed on
excess log-volatility (defined as sj,t − s̄t ). Panel C shows the coefficient when mean log turnover for all
firms in the sample, w̄t , is regressed on mean log volatility, s̄t . Panel D shows the average (time series)
correlations between mean log-volatility and individual company volatility (row 1), and between mean
log-turnover and individual company log-turnover (row 2), for the three groups.
small companies have asymmetric networks, and information about these companies is even
more idiosyncratic.
The correlations between company volatility and turnover are indeed different for companies
of different sizes, as shown in Panel D of Table 3. The panel shows the average (time series)
correlation between market and individual firm log-turnover, Corr(w̄,wj ), and between market
and individual firm volatility, Corr(s̄, sj ), for the three groups. The average correlations are
lower for smaller than for larger companies.
Thus, altogether our tests suggest that price and volume dynamics of small stocks may be
especially well explained by information diffusion through asymmetric networks.
6
Concluding remarks
We have introduced a general network model of a financial market with decentralized information
diffusion, allowing us to study the effects of heterogeneous preferences and asymmetric diffusion
of information among investors, and the interplay between the two. At the individual investor
30
level, our results show that the trading behavior of investors is closely related to their positions in
the network: Closer agents have more positively correlated trades even over short time periods,
and standard network centrality measures are closely related to agents’ profits.
At the aggregate level, the dynamics of a market’s volatility and trading volume is related
to—and could therefore be used to draw inferences about—the underlying information network.
In an initial empirical investigation, we show that the dynamics for large stocks are consistent
with symmetric information networks and large public information components, whereas the dynamics of smaller stocks suggest asymmetric information diffusion through nonpublic channels.
We leave to future research to shed further light on the underlying information network in the
market.
Another future line of research may be the study of endogenous network formation in financial
markets. Given the strong characterization of individual agents’ welfare, the model may be
extended to include a period before signals are received and trading occurs, during which agents
form connections in anticipation of the future value these will generate. It is an open question
what type of networks will form in equilibrium in such an extension, and how well those networks
could match observed dynamics at the individual and aggregate level.
31
Proofs
Proof of Theorem 1:
We prove the result using a slightly more general formulation, where the volatility of noise trade demand is allowed to
vary over time, so instead of τu , we have τu1 , . . . , τuT . We first state three (standard) lemmas.
Lemma 4 (Projection Theorem) Assume a multivariate signal [μ̃x ; μ̃y ] ∼ N ([μx ; μy ], [Σxx , Σxy ; Σyx , Σyy ]). Then the
conditional distribution is
−1
μ̃x |μ̃y ∼ N μx + Σxy Σ−1
yy (μ̃y − μy ), Σxx − Σxy Σyy Σyx .
Lemma 5 (Special case of projection theorem) Assume an K-dimensional multivariate signal v = [v; s] ∼ N (v̄1, σv2 11 +
Λ2 ), where Λ = diag(0, σ1 , . . . , σM −1 ). This is to say that v ∼ N (v̄, σv2 ), si = v + ξi , where ξi ∼ N (0, σi2 )’s are independent
of each other and of v, i = 1, . . . , K − 1. Then the conditional distribution is
v|s ∼ N
Here, τ = (τ1 , . . . , τK−1 )T , τi = σi−2 , τ =
M −1
i=1
1
τv
1
v̄ +
τ s,
τv + τ
τv + τ
τv + τ
.
τi , and τv = σv−2 .
Lemma 6 (Expectation of exponential quadratic form) Assume x ∼ N (μ, Σ), and that B is a symmetric positive
semidefinite matrix. Then
1 E e− 2 (2a x+x Bx) =
−1
−1
−1
−1
−1
1
1
e− 2 (μ Σ μ−(Σ μ−a)(Σ +B) (Σ μ−a)) .
|I + ΣB|1/2
The structure of the proof is now quite straightforward, the extension compared with previous literature being the
heterogeneous information diffusion. We first assume that agents’ demand takes a linear form at each point in time, and
calculate the market maker’s pricing function given observed aggregate demand in (3). This turns out to be linear in a way
such that the market maker’s information is completely revealed in prices. Thus, pt and wt convey the same information.
We then close the loop by verifying that given the market maker’s pricing function in each time period, each agent when
solving their backward induction problem will derive demand and utility according to (5,6), verifying that agents’ demand
functions are indeed linear.
It will be convenient to use the variables Qa,t = τ Va,t . We enumerate the agents from one-dimensionally from 1 to
N̄ , so that agent 1, . . . , N represents the agents in the first replica network, agents N + 1, . . . , 2N , the agents in the second
replica network, etc. Assume that agent a’s time-t demand function is
xa,t (za,t , pt ) = Aa,t za,t + ηa,t (pt ).
Then the total average agent demand is
xt (v, pt ) =
N̄
1 Aa,t za,t + ηa,t (pt ) = At v + ηt (pt ),
N̄ a=1
1 N
1 N
where At = N
a=1 Aa,t , and ηt = N
a=1 ηa,t (pt ), with the convention, A0 = 0, η0 ≡ 0. Here, we are using the fact
1 M −1
that in our large network limM →∞ M
r=0 za+rM,t = v for all a and t (almost surely). The net demand at time t is
then the difference between time t and t − 1 demands,
Δxt = xt (v, pt ) − xt−1 (v, pt−1 ) = (At − At−1 )v + η(pt ) − η(pt−1 ).
Now, the market maker observes total time t demands,
wt = Δxt + ut ,
and since the functions ηt and ηt−1 are known, the market maker can back out
Rt = (At − At−1 )v + ut .
(24)
This leads to the following pricing formula, which immediately follows from Lemma 2.
Lemma 7 Given the above assumptions, the time-t price is given by
pt =
t
τv
τ̂ut
1
v̄
+
v
+
(As − As−1 )τus us ,
τv + τ̂ut
τv + τ̂ut
τv + τ̂ut s=1
32
(25)
−2
where τ̂ut = ts=1 (As − As−1 )2 τus , τus = σu
s .
Equivalently,
pt = λt Rt + (1 − λt (At − At−1 ))pt−1 ,
where λt =
τut (At −At−1 )
,
t
τv +τ̂u
(26)
and p0 = v̄.
Proof of Lemma 7: At time t, the market maker has observed R1 , . . . , Rt . We define the vector s = (R1 /(A1 −
2 /(A − A
2
A0 ), R2 /(A2 − A1 ), . . . , Rt /(At − At−1 )) , and it is clear that si ∼ N (v̄, σv2 + σu
i
i−1 ) ). It then follows immediately
i
from Lemma 2 that
t
τv
1
1
Ri
2
v|s ∼ N
v̄
+
(A
−
A
)
τ
,
,
u
i
i−1
i
τv + τ̂ut
τv + τ̂ut i=1
Ai − Ai−1 τv + τ̂ut
i.e.,
v = Vt + σVt ξtV ,
where Vt =
τv
t v̄
τv +τ̂u
+
1
t
τv +τ̂u
t
i=1 (Ai
2 =
− Ai−1 )τui Ri , σV
t
1
τVt
(27)
, where τVt = τv + τ̂ut .
So, pt = Vt = E[v|s] takes the given form in the first expression of the lemma. A standard induction argument,
assuming that the second expression is valid up until t − 1, shows that the expression then is also valid for t.
2 ). We have shown
Note that (27) is the posterior distribution of v given the information R1 , . . . , Rt , so v ∼ N (Vt , σV
t
Lemma 7.
Thus, linear demand functions by agents imply linear pricing functions in the market, showing the first part of the
proof. We next move to the demand functions and expected utilities of agent a, given the pricing function of the market
maker. We have (using lemma 1 for the posterior distribution) at time t, the distribution of value given {za,t , pt } is
v|{za,t , pt } ∼ N
τVt
τa,t
1
Vt +
za,t ,
τVt + τa,t
τVt + τa,t
τVt + τa,t
.
The time-T demand of an agent can now be calculated. Since individual agents condition on prices, they also observe Ř,
2 /Ā2 )) leads to
and agent a’s information set is therefore {za , Ř}, which via Lemma 1 (with Λ = diag(0, σi2 , σu
⎛
⎞
⎜
⎟
⎜
⎟
τv
τa
τ̂u
1
⎟
v|{za , Ř} ∼ N ⎜
⎜ τ + τ + τ̂ v̄ + τ + τ + τ̂ za + τ + τ + τ̂ Ř, τ + τ + τ̂ ⎟ .
v
a
u
v
a
u
v
a
u
v
a
u
⎝
⎠
2
σ̂a
μa
At time T , given the behavior of the market maker, the asset’s value, given agent a’s information set is therefore
conditionally normally distributed so given agent a’s CARA utility, the demand for the asset (2) takes the form:
xa,T (za , p) =
E[v|Ia,T ] − pT
,
γa σ2 [v|Ia,T ]
a = 1, . . . , N̄ .
The demand of agent a is therefore
xa,T (za,T , pT )
=
=
=
=
=
It follows that AT =
1
N
N
Na,T
a=1 σ 2 γa
μa − pT
γa σ̂a2
1 τv v̄ + τa za + τ̂u Ř − (τv + τa + τ̂u )p
γa
1
τa
τv v̄ + τa za + τ̂u Ř − 1 +
(τv v̄ + τ̂u Ř)
γa
τv + τ̂u
1
(τa za − τa p)
γa
τa,T za,T − pT .
γa
.
33
(28)
2 ), and z
Since v − pT ∼ N (0, σV
a,T − pT = ζa,T + (v − pT ), where ζa,T is independent of v − pT , it follows that
t
ζa,T |(za,T − pT ) ∼ N ( τ
τV
T
+τa,T
(za,T − pT ),
VT
of zero), given za,T − pT , is then
Ua,T
τV
T
1
+τa,T
). The expected utility of the agent at time T (with time-T wealth
=
−E e−γa xa,T (v−pT ) za,T − pT
−E e−τa,T (za,T −pT )(za,T −pT −ζa,T ) za,T − pT
2
−e−τa,T (za,T −pT ) E e−τa,T (za,T −pT )(−ζa,T ) za,T − pT
=
−e−τa,T (za,T −pT ) e
=
−e
=
=
2
τa,T
Vt +τa,T
−τ
τV
T
VT +τa,T
τa,T (za,T −pT ) τ
(za,T −pT )− 1
τ 2 (za,T −pT )2 τ
2 a,T
1
VT +τa,T
(za,T −pT )2 ((τVt +τa,T −τVt )− 1
τ
2 a,T )
2
τa,T
−e
=
−1
(za,T −pT )2
2 τV +τa,T
t
.
τV
T
, using lemma 3.
It is easy to check that the unconditional expected utility is −E0 [e−γa xa,T (v−pT ) ] = − τ +τ
VT
a,T
t
t
Va,t
t
2
We define Yt = τ̂u =
i=1 yi , where yi = (Ai − Ai−1 ) τui , and recall that Qa,t = τa,t = σ 2 =
i=1 qa,i , where
This shows the result at T .
qa,i =
Va,i −Va,i−1
σ2
=
ΔVa,i
.
σ2
With this notation we have
and −E0 [e−γa xa,T (v−pT ) ] = −
Ua,T
=
xa,T
=
τv +YT
τv +YT +τ Va,T
−1
Q2
a,T
(za,t −pt )2
−e 2 τv +YT +QT
Qa,T
(za,T − pT ),
γa
= −√
1
Ca,T
,
= −Da,T .
We proceed with an induction argument: We show that given that (5,6) is satisfied at time t, then it is satisfied at
time t − 1. As already shown, pt−1 , and za,t−1 sufficiently summarizes agent a’s information at time t − 1 (given the linear
pricing function). From the law of motion, Wa,t = Wa,t−1 + xa,t−1 (pt − pt−1 ), an agent’s optimization at time t − 1 is then
Ua,t
=
arg max −Ea,t−1 e−γa Wa,t−1 −γa xa,t−1 (pt −pt−1 ) Da,t e
xa,t−1
=
arg max −Da,t e−γa Wa,t−1 Ea,t−1 e
xa,t−1
def
=
arg max −Da,t e−γa Wa,t−1 Ea,t−1 e
b
Q2
t
−1
(za,t −pt )2 2 τv +Yt +Qt
z
a,t−1 , pt−1
Q2
t
−γa xa,t−1 (pt −pt−1 )− 1
(za,t −pt )2 2 τv +Yt +Qt
z
Q2
t
−b(pt −pt−1 )− 1
(za,t −pt )2 2 τv +Yt +Qt
z
a,t−1 , pt−1
a,t−1 , pt−1 .
(29)
Thus, we need to calculate the distributions of pt − pt−1 and za,t − pt given za,t−1 and pt−1 . From the signal structure,
we have the following relationship
za,t−1
za,t
=
=
v + ξt−1 ,
v + ξt = v +
ξt−1 ∼ N
0,
1
Qt−1
Qt−1
qt
ξt−1 +
et ,
Qt
Qt
,
(30)
et ∼ N
0,
1
qt
,
(31)
where et and ξt−1 jointly independent and independent of all other variables. In the new notation, from (4), we have
pt
=
t
τv
Yt
1
v̄ +
v+
(As − As−1 )τus us ,
τv + Yt
τv + Yt
τv + Yt s=1
(32)
pt−1
=
t−1
τv
Yt−1
1
v̄ +
v+
(As − As−1 )τus us ,
τv + Yt−1
τv + Yt−1
τv + Yt−1 s=1
(33)
A4
34
so
pt − pt−1
=
τv
τv
−
τv + Yt
τv + Yt−1
v̄ +
Yt
Yt−1
1
−
v+
(At − At−1 )τut ut
τv + Yt
τv + Yt−1
τv + Yt √
y t τu
t−1
1
1
−
(As − As−1 )τus us ,
τv + Yt
τv + Yt−1 s=1
+
t
A1
(34)
B1
and also
za,t − pt
Qt−1
qt
τv
τv
ξt−1 +
et −
v̄ +
v
Qt
Qt
τv + Yt
τv + Yt
=
A2
t−1
1
1
(At − At−1 )τut ut −
(As − As−1 )τus us .
τv + Yt τv + Yt s=1
−
√
(35)
yt τut
This leads to the unconditional distribution:
⎡
⎤
⎛⎡
pt − pt−1
st − p t ⎥
⎜⎢
⎦ ∼ N ⎝⎣
st−1
pt−1
⎢
⎣
⎤ ⎡
ΣXX
0
0 ⎥ ⎢
v̄ ⎦ , ⎣ ΣXY
v̄
ΣXY
⎤⎞
⎥⎟
ΣY Y ⎦⎠ ,
(36)
Here,
⎡
ΣXX
=
⎣
=
B1 Yt−1
A2
yt
yt
A1 A2
2
1
+ (τ +Y
− (τ +Y
2 + B1 Yt−1
2 − τv +Yt
τv
τv
v
t)
v
t)
2
B1 Yt−1
Yt−1
A
A1 A2
yt
yt
1
− (τ +Y
+ τ 2 + (τ +Y
2 − τv +Yt
2 + (τ +Y )2
τv
Qt
v
v
t)
v
t)
v
t
yt
0
(τv +Yt )(τv +Yt−1 )
,
1
1
+ τ +Y
0
Qt
v
t
⎡
ΣY Y
=
⎣
1
τv
⎣
ΣXY
=
=
1
Qt
1
Qt−1
A4
τv
⎡
=
+
1
τv
+
1
Qt−1
Yt−1
τv (τv +Yt−1 )
A1
τv
+
A2
τv
A2
4
τv
+
A4
τv
Yt−1
(τv +Yt−1 )2
Yt−1
τv (τv +Yt−1 )
Yt−1
τv (τv +Yt−1 )
0
⎦
⎤
⎦
⎤
⎦,
A4 A1
B1
+ τ +Y
Yt−1
τv
v
t−1
A2 A4
1
1
−
Y
τv
τv +Yt−1 τv +Yt t−1
yt
(τv +Yt )(τv +Yt−1 )
1
1
+ τ +Y
Qt
v
t
⎤
.
0
We use the projection theorem to write [pt − pt−1 ; za,t − pt ] ∼ N (μ, Σ̂), where μ = ΣXY Σ−1
Y Y [za,t−1 − v̄; pt−1 − v̄], and
Σ̂ = ΣXX − ΣXY Σ−1
Y Y ΣXY . It follows that
⎡
μ=⎣
Qt−1 yt
(τv +Yt )(τv +Qt−1 +Yt−1 )
Qt−1 (τv +Yt−1 )(τv +Qt +Yt )
Qt (τv +Qt−1 +Yt−1 )(τv +Yt )
35
⎤
⎦ (za,t−1 − pt−1 ).
We rewrite (29) as
Ua,t
1 arg max −Da,t e−γa Wa,t−1 E e−ax1 − 2 x Bx ,
=
q
!
where x = [pt − pt−1 ; zt − pt ], B = 0, 0; 0,
Q2
t
τv +Yt +Qt
"
, a = [q; 0], and q = γa xa,t−1 . From Lemma 3, it follows directly
that this maximization problem is equivalent to
Ua,t = arg max
−Da,t e−γa Wa,t−1
|I + Σ̂B|1/2
q
1
e− 2 (μ
Σ̂−1 μ−(Σ̂−1 μ−a)Z(Σ̂−1 μ−a))
,
where Z = (Σ̂−1 + B)−1 . Clearly, the optimal solution is given by
arg max q(ZΣ̂−1 μ)1 −
q
leading to q ∗ =
(ZΣ̂−1 μ)1
.
Z11
It is easy to verify that Σ̂−1 μ =
q∗
1
Z11 q 2 ,
2
Qt Qt−1
[1; 1](za,t−1
Qt −Qt−1
− pt−1 ), and some further algebraic
manipulations shows that indeed
= Qt−1 (za,t−1 − pt−1 ), leading to the stated demand function at t − 1, (5).
Given the form of q, it then follows that
μ Σ̂−1 μ − (Σ̂−1 μ − a)Z(Σ̂−1 μ − a) =
Q2t−1
τv + Qt−1 + +Yt−1
(za,t−1 − pt−1 )2 ,
(37)
leading to the form of the utility stated in the theorem (6), with Ca,t−1 = |I + Σ̂B|1/2 . It is easy to check that Ca,t−1
−1/2
takes the prescribed form, as does then Da,t−1 = Ca,t−1 Da,t .
Thus, given a linear pricing function, agents’ demand take a linear form and, moreover, the coefficients take the
functional forms shown in the Theorem, as do agents’ expected utility. We are done.
Proof of Theorem 2: The proof follows immediately from (5), (30-31), and (35).
Proof of Theorem 3: The certainty equivalent satisfies
−e−γa CE
= E0 [−e
−WT +1
] = −E0
#T
1
Q2
1
−1 − 2 τv +Q1 +Y1 (za,1 −p1 )
t=2 Ct−1 e
It is easy to see that
T
$
t=2
−1
Ct−1
=
T τv + QT + YT
τv + Y1 $
Qt−1 (Yt − Yt−1 )
)
1+
.
τv + Q1 + Y1
τv + YT t=2
(τv + Yt−1 )(τv + Yt )
Moreover, since at t = 0, za,1 − p1 ∼ N (0,
1
Q1
E0 e
+
1
),
τv +Y1
it follows that
Q2
1
−1
(za,1 −p1 )2
2 τv +Q1 +Y1
=
τv + Q1 + Y1
,
τv + Y1
and the result for the ex ante certainty equivalent follows.
The ex ante expected profits between t and t + 1 are E0 [(za,t − pt )(pt+1 − pt )]. Plugging in the form (34,35) yields
the result. Using a similar approach for expected profits, we get that the expected total, time T trading profit of agent a’s
trade in time t is
1 Qa,t
,
γa τv + Yt
36
2
.
and the total expected trading profit over time therefore is
T
1 Qa,t
,
γa t=1 τv + Yt
(38)
We are done.
Proof of Theorem 4:
Proof of Theorem 4 : We focus on the Erdös-Renyi random graph G(N, pN ) model, where
pN
=
c logk (N )/(N − 1),
qN
=
p(N − 1) = c logk (N ).
c > 0,
k > 3,
(39)
(40)
Throughout the proof we will often suppress the dependence of variables on N , e.g., writing p and q instead of pN and
qN . Also, denotes a constant arbitrarily close to zero, and C, c, c1 , etc. denote strictly positive (possibly large) constants.
The proof is based on the fact that for arbitrary constants, α > 0, β > 0, it follows that q α << N β , for large N . We
shall see that the variables of interest can be given as well-behaved functions of q −α with error terms of order N −β , which
then leads to tight bounds on the expected correlations.
We first introduce some convenient notation. For a sequence of random variables wN , we write w = Os (f (N )), where
it is assumed that limN→∞ f (N ) = 0, if for large N
P(|w| ≥ Cf (N )) ≤ Cf (N )
for some constant, C > 0. If wN − r = Os (f (N )), for some function (or constant) r, we also write wN = r + Os (f (N )).
We will also use standard O() (Big-O) and o() (little-o) notation. Finally, aN ∼ bN means that 0 < c ≤ aN /bN ≤ C < ∞
for large N (i.e., that aN = Θ(bN )).
We use the notation V ar(x), Cov(x, y), and Corr(x, y) for the population variance, covariance and correlation, respectively, of random variables x and y, whereas we use V [X], C[X, Y ], and ρ[X, Y ] for the sample (cross sectional) variance,
covariance and correlation of vectors of realization of random variables, X and Y .
We are interested in the expectations of the cross sectional correlations
ρD
ρF
ρĈ
ρK
def
=
def
=
def
=
def
=
ρ[DN , ΠN ],
ρ[−F, ΠN ],
ρ[Ĉ, ΠN ],
ρ[K α , ΠN ].
We begin with deriving properties of yt and ζt = βt /β1 . To do this, we start with the following Lemma that shows the
distributional properties of Va,t for large N .
Lemma 8 Consider constants K > 14 (arbitrarily large), and > 0 (arbitrarily close to zero), and an integer T > 1
(arbitrarily large). Then there is an N0 , such that for all N > N0 , for all a = 1, . . . , N , for all t = 1, . . . , T ,
P Va,t − Va,t−1 − q t ≥ q t ≤ N −K .
(41)
Proof : We apply Lemma 10.7 in Bollobas (2001). Given that αt = (K − 2) log(N )/(c logk (N ))t ), βt = (c logk (N ))t /N ,
and γt = 2(c logk (N ))t−1 /N , it follows that α1 ≤ C log(N )
t
1−k
2
, and
αi + βi + γi = α1 (1 + o(1)),
i=1
where αi , βi and γi are defined in Lemma 10.6 of Bollobas (2001). Lemma 10.7 in Bollobas (2001) now implies that,
P Va,t − Va,t−1 − q t ≥ δt q t ≤ N −K ,
37
(42)
where
δt ≤ C log(N )
1−k
2
,
t = 1, . . . , T,
(43)
and since δt → 0 when N → ∞, the result follows.
Bounds on yt :
We recall that Va,t is the number of nodes within a distance of t from node a, and derive asymptotic properties of the
averages
N
def 1
Va,t ,
t = 1, . . . , T.
(44)
V̄t =
N a=1
We note that At = γτ V̄t , so all asymptotic results we derive for V̄t will, up to a constant, also hold for At , and thereby be
important for yt . We have
yt
α(V̄t − V̄t−1 )2 ,
=
2
where we define V̄0 = 0, and α = τuγτ (the risk aversion coefficient being constant since we assume a preference symmetric
economy). Lemma (8) immediately implies that
P V̄t − V̄t−1 − q t ≥ q t ≤ N −K ,
for any > 0, for large N , and therefore
V̄t − V̄t−1 − q t ≤ q t
=
yt − αq 2t α V̄t − V̄t−1 − q t × V̄t − V̄t−1 + q t ≤
αq t × (1 + )q t
=
q 2t .
⇒
Since is also arbitrarily close to zero, we get:
P yt − αq 2t ≥ q 2t ≤ N −K .
(45)
Bounds on ζt :
We can write the expected profit of agent a as
Πa = β 1
Va,1 +
T
ζt Va,t
,
t=2
where β1 = (τv + y1 )−1 and
ζt =
τv + y1
t
τv +
k=1
yk
.
The coefficients, ζt (which do not depend on a) represent the relative value for agents of having many links at distance t.
The bound on yt (45) now immediately imply that
%
&
P ζt × q 2(t−1) − 1 ≥ ≤ N −K .
(46)
for any > 0, K > 14, for large enough N .
Asymptotic behavior of E[ρD ], and E[ρK ]:
First, we restrict ΩN to only contain events for which ζt q 2(t−1) ∈ [1 − , 1 + ], t = 1, . . . , T , for some fixed << 1.
From (46) we know that this set of events satisfies
P(ΩN ) ≥ 1 − O(q −K )
for arbitrarily large K.
38
(47)
We have
ρ[DN , ΠN ]
=
=
=
N
N
N
N
T N
ρ[Va,1
, β1 (Va,1
+ ζ2 Va,2
+ ζ3 Va,3
+ · · · + ζN
Va,T )]
S12 + ζ2 C[Va,1 , Va,2 ] + · · · + ζT C[Va,1 , Va,2 ]
'
S1 V [Va,1 + ζ2 Va,2 + · · · + ζT Va,T ]
S12 + T
i=2 ζi Si,1
,
T
2
2
2
S1 S1 + i=2 ζi Si + T
i=1,j>i 2ζi ζj Sj,i
(48)
where we have defined ζ1 = 1, and
St2
=
V [Va,t ],
St,s
=
C[Va,t , Va,s ],
t = 1, 2, . . . T,
(49)
s, t = 1, 2, . . . , T.
(50)
We define at = ζi q 2(t−1) , t = 2, . . . , T , where the coefficients ai can be chosen arbitrarily close to 1 by letting N be
sufficiently large and focusing on events in Ω defined in (47), we have good control over the behavior of ζt . We also need
to control St2 , and St,s , t > s.
The following Lemma is helpful
Lemma 9 Given the definitions (49), (50), we have
St2
t
=
Zi2 + 2
i=1
St,s
=
t
t
q j−i Zi + Os (N −1/3+ ),
1 ≤ t ≤ T,
(51)
i=1 j=i+1
Ss2 +
s
t
q j−i Zi + Os (N −1/3+ ),
1 ≤ s, t ≤ T.
(52)
i=1 j=s+1
Here,
Zt2
=
qt
t−1
qi ,
i=0
=
q 2t−1
t−1
q −i ,
t = 1, . . . , T,
i=0
Proof of Lemma 9:
For large N , the distributions of Va,t for small t will resemble those of a branching process, the difference being that
branching processes do not choose from the same nodes when going from t to t + 1. For large N , the difference will be
Va,t
small. We therefore begin with studying the branching process defined as Za,1 = ξa,0 , Za,t+1 = i=1
ξa,t,i , a = 1, . . . , N ,
t = 1, . . . , T − 1. Here, ξa,t,i ∼ Bin(N − 1, p), and are independent across a, t, and i. Using iterated expectations, and the
law of total variance, it is straightforward to show the following (unconditional) moment formulas
E[Za,t ]
=
E[ξ]t ,
V ar[Za,t ]
=
E[ξ]t V ar(ξ) + E[ξ]2 V ar[Za,t−1 ],
Cov[Za,t , Za,s ]
=
E[ξ]t−sV ar(Za,s ),
t > s.
In the specific case where E[ξ] = q, and V ar(ξ) = q + O(N −1 )—i.e., our case—a recursive argument shows that
E[Za,t ]
=
qt ,
V ar[Za,t ]
=
qt
t−1
q i + O(N −1+ ),
(53)
i=0
Cov[Za,t , Za,s ]
=
qt
s−1
q i + O(N −1+ ),
i=0
39
t > s.
(54)
We next show that the moments for Va,t+1 − Va,t are similar to the moments of Za,t+1 . Of course, Va,1 has identical
moments as Za,1 . For Va,2 , we have, given that node a has Va,1 links, the probability that node b which is not a neighbor
of a is not a neighbor to one of a’s neighbors is (1 − p)Va,1 . Therefore, the conditional distribution of the number of nodes
at distance 2 from a, given Va,1 is
Va,2 − Va,1 ∼ Bin(N − Va,1 , 1 − (1 − p)Va,1 ),
leading to
E[Va,2 − Va,1 |Va,1 ]
=
=
=
(N − Va,1 )(1 − (1 − p)Va,1 )
⎞
⎛
Va,1 %
Va,1 &
−N (1 − Va,1 /N ) ⎝
(−1)i pi ⎠
i
i=1
⎛
⎞
Va,1 %
Va,1 & −q i
⎝
⎠,
−N (1 − Va,1 /N )
i
N −1
i=1
(55)
and
V ar[Va,2 − Va,1 |Va,1 ]
=
=
=
(N − Va,1 )(1 − (1 − p)Va,1 )(1 − p)Va,1
⎛
⎞
Va,1 %
Va,1 &
Va,1 ⎝
i i⎠
−N (1 − Va,1 /N )(1 − p)
(−1) p
i
i=1
⎛
⎞
Va,1 %
Va,1 & −q i
⎠.
−N (1 − Va,1 /N )(1 − p)Va,1 ⎝
i
N −1
i=1
(56)
The law of total variance now implies that
V ar[Va,2 − Va,1 ]
E[V ar[Va,2 − Va,1 |Va,1 ]] + V ar[E[Va,2 − Va,1 |Va,1 ]],
=
i /(N −1)] = O(N −1+ ) for any fixed i and t, and furthermore, E
and since (as is easily shown) E[Va,t
∞
O(N −1 ), (55,56) immediately imply, when plugged into (57), that
V ar[Va,2 − Va,1 ]
(57)
i=2
i /(N − 1)i =
Va,t
q 2 + q 3 + O(N −1+ ) = V ar[Za,1 ] + O(N −1+ ).
=
The same argument for
Va,3 − Va,2 ∼ Bin(N − (Va,2 − Va,1 ), 1 − (1 − p)Va,2 −Va,1 ),
leads to
V ar[Va,3 − Va,2 ]
=
q 3 + q 2 (q 2 + q 3 ) + O(N −1+ ) = V ar[Za,2 ] + O(N −1+ ),
and for higher orders to
V ar[Va,t − Va,t−1 ] = V ar[Za,t ] + O(N −1+ ).
(58)
A similar argument for the covariances leads to,
Cov[Va,t − Va,t−1 , Va,s − Va,s−1 ]
=
Cov[Za,t , Za,s ] + O(N −1+ ),
t > s.
(59)
Now, since Va,t = Va,1 + (Va,2 − Va,1 ) + . . . + (Va,t − Va,t−1 ), from (58,59) it immediately follows that
V ar[Va,t ]
=
t
i=1
Zi2 + 2
t
t
i=1 j=i+1
40
q j−i Zi + Os (N −1+ ),
and
(60)
Cov[Va,t , Va,s ]
=
V ar[Va,s ] +
s
t
q j−i Zi + Os (N −1+ ),
t > s.
(61)
i=1 j=s+1
m − 1, is of course
We next show that covariances of Va,t and Vb,t when a = b (i.e., across agents) is low. First, Va,1
binomially distributed,
Va,1 − 1 ∼ Bin(N, p).
We recall that the moment generating function of a binomial distribution is M (t) = (1 − p + pet )n , and it therefore follows
that for any fixed m,
μm
m
]
V ar[Va,1
=
m
E[Va,1
] = M (m) (0) = q m (1 + O(q −1 )),
=
M (2m) (0) − (M (m) (0))2 = q 2m−1 (1 + O(q −1 )).
def
m and V n , a = b is on the form
The covariance between Va,1
b,1
m
n
Cov(Va,1
, Vb,1
) = E[(wa + I)m (wb + I)n ] − E[(wa + I)m ]E[(wb + I)n ],
where I is Bernoulli distributed, I ∼ Ber(p), representing the possible common link between a and b, wa ∼ Bin(N − 1, p),
wb ∼ Bin(N − 1, p) represent all the other links, and I, wa , and wb are independent.
Moreover, since the Bernoulli distribution is idempotent, I k = I, k ≥ 1, it follows that
m
n
, Vb,1
)
Cov(Va,1
=
=
−
=
=
=
E[(wa + I)m (wb + I)n ] − E[(wa + I)m ]E[(wb + I)n ]
⎡
⎞⎤
⎛
m−1
n−1
%m&
%n& j
m
i
n
E ⎣ wa +
wa I ⎝wb +
w b I ⎠⎦
i
j
i=0
j=0
⎤
⎡
m−1
n−1
%m&
%n& j
E wam +
wai I E ⎣wbn +
wb I ⎦
i
j
i=0
j=0
⎡
⎞⎤
⎤
⎛n−1
m−1
⎡n−1
m−1
%m&
%n& j
%m&
%n& j
i
i
⎣
⎝
⎠
⎦
⎣
wa I
wb I
wa I E
wb I ⎦
E
−E
i
j
i
j
i=0
j=0
i=0
j=0
⎞
⎞
m−1
m−1
⎛n−1
⎛n−1
%m&
%n&
%m&
%n&
⎠
⎝
⎝
μi
μj E[I] −
μi
μj ⎠ E[I]2
i
j
i
j
i=0
j=0
i=0
j=0
⎞
⎛n−1
m−1
%m&
%n&
⎝
μi
μj ⎠ V ar(I)
i
j
i=0
j=0
=
O(q m+n−2 p(1 − p))
=
O(N −1+ ).
m ), this then implies that
Together with the bounds on V ar(Va,1
m
n
Corr(Va,1
, Vb,1
)
=
m ,V n )
Cov(Va,1
b,1
'
V ar(Va,1 )m V ar(Vb,1 )n
=
O(N −1+ )
'
2m−1
q
q 2n−1 (1 + O(q −1 ))
=
O(N −1+ ).
An identical argument as that above shows that
Cov((Va,1 − Vb,1 )m , (Vc,1 − Vd,1 )n )
=
O(q m+n−2 N −1+ ) = O(N −1+ ),
when a, b, c and d are different, and that
Corr((Va,1 − Vb,1 )m , (Vc,1 − Vd,1 )n )
41
=
O(N −1+ ),
and similarly for general T ≥ t ≥ s ≥ 1, n ≥ 1, m ≥ 1
m
n
, Vb,s
)
Cov(Va,t
=
O(N −1+ ),
−1+
(62)
m
n
Corr(Va,t
, Vb,s
)
=
O(N
),
(63)
Cov((Va,t − Vb,t )m , (Vc,s − Vd,s )n )
=
O(N −1+ ),
(64)
Corr((Va,t − Vb,t ) , (Vc,s − Vd,s ) )
m
n
=
O(N
−1+
).
(65)
To go from these bounds on population variances and covariances to bounds on sample variances and covariances, we
use the following lemmas:
Lemma 10 Assume x1 , . . . , xN are identically distributed (but not necessarily independent) random variables, with mean
μ and (finite) variance σ2 . Also, assume that Corr(xi , xj ) ≤ CN −α , α > 0. Then the sample mean
def
x̄N =
satisfies
N
1 xi
N i=1
x̄N = μ + Os (max(σ, 1)N −β/3 ),
where β = min(α, 1). Here, μ and σ are allowed to depend on N . Moreover, if σ is independent of N , the formula reduces
to
x̄N = μ + Os (N −β/3 ).
σ2
2 2 −β . Chebyshev’s inequality then
Proof of Lemma 10: It is clear that V ar(x̄N ) = N
2 (N +
i,j=i Corr(xi , xj )) ≤ C σ N
immediately implies that
P(|x̄N − μ| ≥ CσN −β/2 K) ≤ K −2 ,
and by choosing K = N β/6 the result follows.
Lemma 11 If
x̄N = μ + Os (N −α+ )
for all > 0, then, for any fixed m > 1, such that μm−1 = o(N ) for all > 0,
m
−α+
).
x̄m
N = μ + Os (N
Proof of Lemma 11: The proof is by induction. Assume that the result has been proved for all m = 1, 2, . . . , M , i.e.,
m
−α+
) ≤ Cm N −α+ ,
P(|x̄m
N − μ | ≥ Cm N
We expand
m = 1, 2, . . . , M.
M
+1
M +1
i M −i −
μ
|
=
|x̄
−
μ|
a
μ
x̄
|x̄M
,
i
N
N
N
i=0
for some constants ai , i = 1, . . . , M . Now, clearly under our induction assumption,
m
−α+
,
P(|x̄m
N | ≥ 2μ ) ≤ N
for large enough N . Therefore,
m = 1, . . . , M
M
i M −i M
ai μ x̄N ≥ C μ
P ≤ C N −α+ ,
i=0
for some
C
> 0. Moreover,
P(|x̄N − μ| ≥ C1 N −α+ ) ≤ C1 N −α .
i M −i ≤ C μM and |x̄ − μ| ≤ C N −α , then
and since if M
1
N
i=0 ai μ x̄N
+1
|x̄M
N
−μ
M +1
| = |x̄N
M
i M −i − μ| ai μ x̄N ≤ C1 N −α+ C μM ≤ C N −α+2 .
i=0
42
Finally,
M
i M −i M
−α
≥ 1 − C1 + C N −α = 1 − C N −α .
ai μ x̄N ≤ C μ ∩ |x̄N − μ| ≤ C1 N
P i=0
Setting CM +1 = max(C , C ), and recalling that > 0 was arbitrary, the lemma follows.
Lemma 12 Assume x1 , . . . , xN are identically distributed (but not necessarily independent) random variables, with mean
μ, variance σ2 , and finite fourth moments. Assume that |Corr(xi , xj )| ≤ CN −α and that |Corr((xi − xj )2 , (xk − xn )2 )| ≤
CN −α , where α > 0, and i, j, k, n are different indexes. Then the sample variance
N
1 (xi − x̄N )2
N − 1 i=1
def
s2N =
satisfies
s2N = σ2 + Os (N −2β/3 ),
where β = min(α, 1).
Proof of Lemma 12: We rewrite
def
s2N =
We have
1
1
(xi − xj )2 .
N (N − 1) i,j 2
E[(xi − xj )2 ] = E[((xi − μ) − (xj − μ))2 ] = V ar(xi ) + V ar(xj ) − 2Cov(xi , xj ),
implying that
def
E[s2N ] = σ2 − Cov(xi , xj ) = σ2 (1 − r) = σ̄2 ,
in turn leading to
V
ar(s2N )
=
1
N (N − 1)
2
r = O(N −α ),
⎡⎛
⎞ ⎤
2
1
2
2 ⎠ ⎦
⎣
⎝
E
(xi − xj ) − σ̄
.
2
i,j
We expand the square, and separate
• N (N − 1) terms on the form:
1
N (N − 1)
2
E
1
(xi − xj )2 − σ̄2
2
2 .
The sum of these terms will thus be of O(N −2 ).
• N (N − 1)(N − 2) terms on the form:
1
N (N − 1)
!
2
E
1
(xi − xj )2 − σ̄2
2
1
(xi − xk )2 − σ̄2
2
"
.
The sum of these terms will thus be of O(N −1 ).
• N 4 − 3N 3 − N 2 + N terms on the form:
1
N (N − 1)
!
2
E
1
(xi − xj )2 − σ̄2
2
1
(xk − xn )2 − σ̄2
2
"
.
Since
!
"
1
1
2
2
2
2
E
−
x
)
−
σ̄
−
x
)
−
σ̄
(x
(x
n
i
j
k
2
2
=
×
=
the sum of these terms will be of O(N −α ).
43
Corr 1 (xi − xj )2 , 1 (xk − xn )2 2
2
1
(xi − xj )2
V ar
2
O(N −α ),
Thus, altogether, V ar(s2N ) = O(N −β ). As in Lemma 10, the result then follows from Chebyshev’s inequality.
Lemma 13 Assume x1 , . . . , xN , and y1 , . . . , yN , are identically distributed random variables, with means μX , μY , vari2 , σ 2 , covariance σ
−α ,
ances σX
XY = Cov(xi , yi ) for all i, and finite fourth moments. Assume that |Corr(xi , yj )| ≤ CN
Y
2
2
−α
and that |Corr((xi − xj ) , (yk − yn ) )| ≤ CN
, where α > 0, and i, j, k, n are different indexes. Then the sample
covariance
N
1 def
sXY =
(xi − x̄N )(yi − ȳN )
N − 1 i=1
satisfies
sXY = σXY + Os (N −2β/3 ),
where β = min(α, 1).
Proof of Lemma 13: Identical to the Proof of Lemma 12.
Lemma 12, together with (60) and (63) therefore implies (51), and Lemma 13 together with (61) and (65) implies (52).
.
We have shown Lemma 9.
Plugging (51-52) into (48), we get
ρ[DN , ΠN ]
=
S1
=
q(1
S12 + T
i=2 ζi Si,1
+ Os (N −1/3+ )
T
2
2
2
S1 + i=2 ζi Si + T
i=1,j>i 2ζi ζj Sj,i
+ a22 (q −2
−1/3+
+
q(1 + a2 (q −1 + q −2 ) + a3 (q −2 + q −3 ) + a4 q −3 + q −4 P1 (q −1 ))
+ 2a2 (q −1 + q −2 ) + 2a3 (q −2 + q −3 ) + 2a4 q −3 + 2a2 a3 q −3 + q −4 Q(q −1 ))1/2
3q −3 )
+
Os (N
),
=
1 + a2 (q −1 + q −2 ) + a3 (q −2 + q −3 ) + a4 q −3 + q −4 P1 (q −1 )
1 + a22 (q −2 + 3q −3 ) + 2a2 (q −1 + q −2 ) + 2a3 (q −2 + q −3 ) + 2a4 q −3 + 2a2 a3 q −3 + q −4 Q(q −1 )1/2
+
Os (N −1/3+ ),
(66)
where P and Q are polynomials finite orders, and we define a1 = 1. A Taylor expansion of (66) around x = 0, where
x = q −1 now yields,
1+
a22 (q −2
+
1 + a2 (q −1 + q −2 ) + a3 (q −2 + q −3 ) + a4 q −3 + q −4 P1 (q −1 )
= 1 − cq −3 + O(q −4 )
+ 2a2 (q −1 + q −2 ) + 2a3 (q −2 + q −3 ) + 2a4 q −3 + 2a2 a3 q −3 + q −4 Q(q −1 )1/2
3q −3 )
altogether leading to
ρD
=
1 − cq −3 + Os (q −4 ),
(67)
where c > 0 depends on a2 , . . . , aT , and when at → 1, i = 2 . . . , aT , then c → 1/2. Of course, this immediately implies that
P |1 − ρD − cq −3 | ≥ Cq −4 ≤ Cq −4 ,
(68)
for large N for some bounded constant C > 0.
Now, since correlations are bounded between -1 and 1, for general sequences of random variables, aN , and bN , if
P (|1 − ρ[aN , bN ] − f (N )| ≥ o(f (N ))) = o(f (N )),
where limN→∞ f (N ) = 0, then
Thus, from (68), it follows that
1 − E(ρ[aN , bN ]) = f (N )(1 + o(1)).
1 − E[ρD ] = cq −3 (1 + o(1)) ∼ q −3 .
A similar expansion of VN1 + ζ2 VN2 gives
ρ[Va,1 + ζ2 Va,2 , ΠN ]
=
44
1 − cq −5 + Os (q −6 ),
(69)
where c = 1/2 when ai → 1, i = 2 . . . , aT , which then leads to
1 − E[ρK ] ∼ q −5 .
(70)
This establishes the relationship E[ρK ] > E[ρD ] for large N .
We next study F = 1 , the average distance from an agent to all other agents, and show that
Ĉ
1 − E[ρ[D, F ]] ∼ q −1 .
(71)
We note that (53,54) imply that for i > 1, and fixed α,
Corr(Z1 , Z1 + αZi )
1 ,Zi )
1 + α Cov(Z
V ar(Z1 )
Cov(Z1 ,Zi )
V ar(Z )
1 + 2α V ar(Z
+ α2 V ar(Z i )
)
=
1
1
1 + αq i−1
'
1 + 2αq i−1 + α2 (q 2i−2 + q 2i−3 + O(q 2i−4 ))
=
1 + (αq)−1
'
1 + 2(αq)−1 + q −1 + O(q −2 ))
1
1 − q −1 + O(q −2 ),
2
=
=
in turn implying that for α1 , . . . , αR , such that α1 /( i αi ) ≤ 1 − , > 0,
Corr
Z1 ,
R
=
Zi
1−
i=1
1 −1
q
+ O(q −2 ).
2
(72)
Now, from Theorem 10.10 in Bollobas (2001), it follows that if we define R = log(N )/ log(c) + 1, with probability greater
than 1 − N −k for arbitrary k, the maximum distance between any two agents is either R − 1 or R. We will therefore have
that the average distance between an agent, a, and all other agents is
Fa
=
=
1
N
R−
Va,1 +
R−1
i(Va,i − Va,i−1 ) + R
N − (Va,1 +
R−1
i=2
R−1
Va,1 −
N
(Va,i − Va,i−1 )
i=2
R−1
i=2
R−i
(Va,i − Va,i−1 ).
N
Now, from (72), if we replace Va,i − Va,i−1 with Za,i ,
F̃a
def
=
=
1
N
R−
Za,1 +
R−1
(iZi ) + R
N−
Za,1 +
R−1
i=2
Z1
i=2
R−1
R−i
R−1
Za,1 −
Za,i
N
N
i=2
we have
Corr(Va,1 , −F̃a ) = 1 −
1 −1
+ O(q −2 ).
q
2
A similar argument as that leading to (67) therefore implies that
Corr(Va,1 , −Fa ) = 1 −
1 −1
+ O(q −2 ),
q
2
and
ρ[D, −F ] = 1 −
q
+ Os (q −2 ),
2
45
(73)
and therefore, following a similar argument as for ρD , that
E[ρ[D, −F ]] = 1 −
q
+ O(q −2 ).
2
We use the following triangle-inequality like lemma:
√
√
Lemma 14 Assume ρ(a, b) = 1 − α, and ρ(a, c) = 1 − β, with β > α. Then ρ(b, c) ≤ 1 − ( β − α)2 .
Proof of Lemma 14: Because of the simple renormalizations, a → (a − E[a])/σ(a), b → (b − E[b])/σ(b), and c → (c −
E[c])/σ(c), we can without loss of generality assume that a, b, and c have zero expectations
and unit variances. All
'
correlations can then expressed as, ρ(a, b) = E[ab], etc. We introduce the metric d(a, b) = E[(a − b)2 ], and we then have
the triangle inequality d(a, c) ≤ d(a, b) + d(b, c), leading to d(b, c) ≥ d(a, c) − d(b, c). We have d(a, b)2 = E[(a − b)2 ] =
E[a2 ] + E[b2] − 2E[ab] = 2 − 2(1 − α) = 2α,√and similarly d(a, c)2 = 2β. Finally, d(a, c)2 = 2 − 2ρ(a, c) which via the triangle
√
√
√
inequality leads to 2 − 2ρ(a, c) ≥ ( 2β − 2α)2 = 2( β − α)2 , leading to the result. We are done.
Equations (68) and (73), together with Lemma 14, where a = D, b = −F , c = Π, now implies that if 1−ρ[D, Π] ≤ cq −3 ,
and 1 − ρ[D, −F ] ≥ cq −1 , then
1 − ρ[−F, Π] = 1 − ρF > c(1 − )q −1
for large N , and thus,
1 − E[ρF ] ≥ c(2 − )q −1 ,
implying the third part of the Theorem, E[ρD ] > E[ρF ].
Finally, for Ĉ, we first note that
and more generally that
E[Va,1 ]
=
q + O(N −1+ ),
V ar[Va,1 ]
=
q + O(N −1+ ),
E[Fa ]
=
R(1 + O(q −1 )),
V ar[Fa ]
=
R2
(1 + O(q −1 )),
q
E[(1 − F/R)i ] = q −i (ci + O(q −1 ))
for any fixed i ≥ 1, where ci is a (finite) constant, and c2 = 1. We have
Corr(Va,1 , Ĉ)
=
=
=
=
=
=
1
Corr Va,1 ,
F
1
Corr Va,1 ,
R − (R − F )
⎞
⎛
1
&⎠
%
Corr ⎝Va,1 ,
1− 1− F
R
⎞
⎛
F
F i⎠
i
⎝
(−1) 1 −
Corr Va,1 , − +
R i≥2
R
q
di q −i
1 − + O(q −2 ) +
2
i≥2
1−
q
+ O(q −2 ).
2
A similar argument as for ρF , again using Lemma 14, then implies the final part of the theorem, E[ρD ] > E[ρĈ ]. We are
done.
Proof of Theorem 5: For the first part, we note that since (pt −pt−1 ) is independent of pt−1 (given publicly available inforyt
2 = (Σ
mation), it follows that the price volatility between t−1 and t is equal to (ΣXX )11 , σp,t
.
XX )11 |t−1 = (τ +Y )(τ +Y
)
2
with the convention Y0 = 0. Also, the final period volatility of v − pT is σp,T
+1 =
theorem.
46
v
1
τv +YT
t
v
t−1
. This proves the first part of the
t
k1
For the second part, we first note that from the definition of y1 , it follows that y1 = τv 1−k
1
and—defining Kt =
1−KT
kt
i=1 ki —a simple induction argument further shows yt = τv (1−Kt−1 )(1−Kt ) , t = 1, . . . , T − 1, and yT = τv kT (1−KT −1 ) .
We note that all the y1 , . . . , yT are all well defined.
y1
Next, we back out the connectedness that is needed to be consistent with the y’s. We have A1 =
, At =
τu
def
V
def
yt
γ
y
γ
y
a,1
a
At−1 + τ , leading to V̄1 =
= τ τ 1 , ΔV̄t = V̄t − V̄t−1 = τ τ t . Thus, if we can replicate, arbitrarily closely,
N
u
u
u
any sequence of diffusions, through which the average number of signals, V̄t , increases over time, then we can generate any
yt , and thereby any volatility structures. We note that γ is a free parameter that allows us to scale the network to arbitrary
sizes. The result now follows from the following lemma:
Lemma 15 For any T , there are networks of size N , such that V̄T = (1 + o(1))V̄T for T > T , and V̄T =
for T < T .
1
V̄
N(1+o(1)) T
This lemma thus states that we can always find a (possibly large) network such that very little happens before and
after time T , with respect to information diffusion. The result follows immediately: For T = 1, a tightly-knit network would
have these properties. For T = 2, a large star network. For T = 3, a star-like network with N 2 + N nodes, in which there
are N tightly-nit nodes in the center, each connected to N peripheral agents. For even T ≥ 4, adding longer distance to
the T = 2 (star) network, and for odd T ≥ 5, adding longer distances to the T = 3 network will generate these properties.
Let’s call such a network a T -network.
V̄t+1
, t = 1, . . . , T can be generated by choosing a network with many disjoint 1−, 2−, . . . , T −networks
Finally, any sequence of V̄
t
in such a way so that the relative sizes of the networks match the fractions.
We are done.
Proof of Lemma 3:
i): Let us study the function g, defined by gi = fi , i = 0, . . . , N , and gN+1 = 0, which has the same autocorrelation
function as f , for τ ≤ N , and which always has a maximum at an interior point when f is unimodal, nonnegative, and
f0 = 0. Let us denote that maximum by m, i.e., gm ≥ max(gm−1 , gm+1 ). Of course, the result follows trivially if f ≡ 0, so
we assume that fi > 0 for some i.
We use the following lemma
Lemma 16 Consider a sequence f0 , f1 , . . . , fN+1 , such that f0 = fN+1 = 0. Then
N
fi (fi+1 − fi ) − (fk+1 − fk )(fk − fk−1 ) ≤ 0
i=0
for any 1 ≤ k ≤ N .
N
N
Proof of Lemma 16: The summation by parts rule implies that
i=0 fi (fi+1 − fi ) = −
i=0 fi+1 (fi+1 − fi ), in turn
N
2
implying that i=0 fi (fi+1 − fi ) = − 21 N
i=0 (fi+1 − fi ) . We now have
N
fi (fi+1 − fi ) = −
i=0
N
1
(fi+1 − fi )2
2 i=0
≤
1
1
− ((fk − fk−1 )2 + (fk+1 − fk ))2 ) ≤ 2(fk+1 − fk )(fk − fk−1 ),
2
2
where the second inequality follows from the inequality (x + y)2 = x2 + y 2 + 2xy ≥ 0. Thus,
fk )(fk − fk−1 ) ≤ 0, as claimed. This completes the proof of Lemma 16.
N
i=0 fi (fi+1
− fi ) + (fk+1 −
Since Rτ is symmetric around τ = 0, we restrict our attention to the region τ ≥ 0, showing that Rτ is nonincreasing
in this region. This will immediately imply i). We prove the result by contradiction: Assume that there is a τ , such that
Rτ +1 − Rτ > 0. Given the definition of Rτ , this means that
N+1−τ
gi (gi+τ − gi ) > 0
(74)
i=0
for some τ . By zero padding (adding extra zeros to) g, we can without loss of generality assume that the maximum occurs
at m = nτ for some integer n, and that N + 1 − τ = M τ − 1, i.e., by studying the function gi = 0, i < v, gi = gi−v ,
0 ≤ i − v ≤ N + 1 − τ , gi = 0, N + 1 − τ < i − v ≤ M τ − 1. Thus, it must be that
⎛
⎝
(m−1)τ −1
i=0
gi (gi+τ
−
⎞
⎛
gi )⎠
+⎝
mτ
−1
⎞
gi (gi+τ
−
gi )⎠
+
M τ −1
i=mτ
i=(m−1)τ
47
gi (gi+τ
− gi )
> 0.
Now, because gi is increasing for i ≤ mτ , and decreasing for i ≥ mτ , we can replace gi with gr i τ , where ri = τ (i/τ + 1)
in the first and third term, and by g(m−1)τ
for the middle term, leading to
m−2
giτ
τ
−1
(giτ
+j+τ − giτ +j )
i=0
(m−1)τ −1
j=0
g(m−1)τ
τ
−1
M
−1
giτ
(gmτ
+j+τ − gmτ +j )
≥
τ −1
j=0
mτ
−1
gi (gi+τ
− gi ),
i=(m−1)τ
τ
−1
(giτ
+j+τ − giτ +j )
i=m
gi (gi+τ
− gi ),
i=0
j=0
But since
≥
>
j=0
M
τ −1
gi (gi+τ
− gi ).
i=mτ
gk+j+1 − gk+j = gk+τ − gk , this implies that
m−2
giτ
(g(i+1)τ
−
giτ
)
%
&
− g(m−1)τ
(gmτ
− g(m+1)τ
) +
M −1
i=0
in turn leading to
giτ
(g(i+1)τ
−
giτ
)
> 0,
i=m
M −1
giτ
(g(i+1)τ
−
giτ
)
+ (g(m−1)τ
− gmτ
)(g(m+1)τ
− gmτ
) > 0.
i=0
, 0 ≤ i ≤ M , it follows that h = h
Now, defining hi = giτ
0
M = 0, that h is positive and unimodal, and that this in turn
implies that
M −1
hi (hi+1 − hi ) − (hm − hm−1 )(hm+1 − hm ) > 0.
i=0
However, from Lemma 16 no such sequence can exist, and thus neither can a sequence g satisfying (74). So, R is nonincreasing
on the positive axis, and it must therefore be unimodal (with maximum at 0). We have shown i).
n
n
ii): It is easy to show that (Δn R)(τ ) = N−τ
i=1 fi (Δ f )i . Thus, if Δ f is positive (negative), so is Rτ . This, in turn,
implies that Rτ has at most n − 1 turning points.
Proof of Theorem 6: The proof is based on the following standard lemma:
Lemma 17 Assume a normally distributed random variable, y ∼ N (μ, σ2 ). Then E[|y|] = σ
where Φ is the cumulative normal distribution of a standard normal variable.
μ2
2 − 2σ2
e
π
+μ(1−2Φ(−μ/σ)),
We note that from (5) and given that v = v̄ + η, it follows that agent a’s net time-t demand is
γa Δxa,t
=
=
−
=
Qa,t (za,t − pt ) − Qa,t (za,t−1 − pt−1 )
t
qt
τv
Yt
1
Qt−1
ξa,t−1 +
ea,t −
v̄ +
(v̄ + η) +
(Ai − Ai−1 )τui ui
Qa,t v̄ + η +
Qa,t
Qa,t
τv + Yt
τv + Yt
τv + Yt i=1
t−1
τv
Yt−1
1
Qa,t−1 v̄ + η + ξa,t−1 + −
v̄ +
(v̄ + η) +
(Ai − Ai−1 )τui ui
τv + Yt−1
τv + Yt−1
τv + Yt−1 i=1
t−1
Qa,t−1
Qa,t
Qa,t
qa,t ea,t −
−
(Ai − Ai−1 )τui ui −
(At − At−1 )τui ut
τv η −
τv + Yt
τv + Yt−1
τv + Yt
i=1
∼N(0,qa,t )
2 )
∼N(0,r̂a,t
where
r̂t2 =
Qa,t−1
Qa,t
−
τv + Yt
τv + Yt−1
2
(τv + Yt−1 ) +
Qa,t
τv + Yt
48
2
yt =
Q2a,t−1
τv + Yt−1
−
Q2a,t
τv + Yt
+
2
qa,t
τv + Yt
.
Recalling that qa,t = τ ΔVa,t and Qa,t = τ Va,t , this leads to
γa Δxa,t =
'
τ ΔVa,t
√
ra,t τ
ωa,t + '
ξt ,
ΔVa,t
where ωa,t ∼ N (0, 1) is independent of ξt ∼ N (0, 1) and across agents,
rt2 =
2
Va,t−1
τv + Yt−1
when ΔVa,t > 0, and
−
2
Va,t
τv + Yt
+
2
ΔVa,t
τv + Yt
,
γa Δxa,t = ra,t τ ξt ,
√
Δx
r
τ
when ΔVa,t = 0. Assuming that ΔVa,t > 0, we note that γa τ ΔVa,t ξt ∼ N √a,t
ξt , 1 . The law of large numbers,
a,t
ΔV
a,t
together with Lemma 17, then in turn implies that, conditioned on ξt ,
⎛
⎞
2 ξ2
√
√
τ ra,t
t
'
− 2ΔV
1 2
τ
τ
r
r
a,t
a,t
a,t + '
⎠.
e
|Δxa,t | →a.s. τ ΔVa,t ⎝
ξt 1 − 2Φ − '
ξt
γa
M a
π
ΔVa,t
ΔVa,t
It follows immediately that the unconditional expectation of the first term (not conditioning on ξt ) is
√
2 τ ΔVa,t
.
π
2 +ΔV
τ ra,t
a,t
For the second term, we use the fact that E[yΦ(ay)] =
√
2 τ ΔVa,t
π
2
τ ra,t
+1
ΔV
1
2π
√
a
,
a2 +1
for a random variable y ∼ N (0, 1),
to get
2
ra,t
τ
√
√
2
'
τ ra,t
2'
2√
ra,t τ
ra,t τ
ΔVa,t
=
τ ΔVa,t E '
ξt 1 − 2Φ − '
ξt
τ ΔVa,t =
τ
.
2
π
π
2
ΔVa,t
ΔVa,t
ra,t τ
τ ra,t + ΔVa,t
1 + ΔV
a,t
Summing the two terms together, we get
( 1 τ
2
2 + ΔVa,t .
E
|Δxa,t | →a.s.
ra,t
M a
γa π
τ
We note that this formula also holds when ΔVa,t = 0, since E|ra,t τ ξt | = τ
Xt
=
=
=
N
τ 1
N a=1 γa
(
2
π
2 +
ra,t
ΔVa,t
τ
2 2
r .
π a,t
This finally leads to (20)
) *
N
2
Va,t−1
yt
τ 1 *
ΔVa,t
2Va,t−1 ΔVa,t
+2
+
−
N a=1 γa π (τv + Yt−1 )(τv + Yt )
τv + Yt
τ
) *
N
2
2
2
Va,t−1
Va,t
ΔVa,t
τ 1 *
ΔVa,t
+2
.
−
+
+
N a=1 γa π τv + Yt−1
τv + Yt
τv + Yt
τ
Now, if one of the γa → 0, then agent Va,t will determine At , yt , and Yt . In this case, we get
Xt
=
) ⎛
*
*
N
τ 1 *2 ⎝
+
N a=1 γa π (τv +
2
2 τu τ ΔV 2
Vt−1
t
γ2
τu τ 2
2
γa
t−1
i=1
a
ΔVt2 )(τv +
49
τu τ 2
2
γa
t
i=1
=
a,t
ΔVt2 )
−
τv +
2Vt−1 ΔVt
τu τ 2 t
2
γa
i=1
ΔVt2
⎞
ΔVt ⎠
+
.
τ
Using the fact that Vt =
ΔVt , leading to the inequality Vt2 ≤ t ti=1 ΔVi2 (from E[x2 ] ≥ E[x]2 ), it follows that for large
ΔVt , the third term will be dominant, and therefore any sequence of Xt can be generated by choosing ΔVt appropriately.
This shows the second part of the theorem.
We are done.
Proof of Theorem 7: We first show the second result: We use (24,26) to define
R̂t = (At − At−1 )τut Rt − yt v̄ = yt η + (At − At−1 )τui ui ,
and, of course, R̂1 , . . . , R̂t can be backed out of p1 , . . . , pt . Now, it is straightforward to use (26) to show that
pt
=
v̄ +
=
v̄ +
1
τv + Yt
1
τv + Yt
Yt η +
t
(Ai − Ai−1 )τui ui
i=1
t
R̂t = v̄ +
i=1
1
τv + Yt
Yt η +
t
(Ai − Ai−1 )τui ui
.
(75)
i=1
Using this relation, we get
γa Δxa,t
=
=
−
=
−
=
Qa,t (za,t − pt ) − Qa,t−1 (za,t−1 − pt−1 )
t
qt
1
Qt−1
R̂t
ξa,t−1 +
ea,t − v̄ +
Qa,t v̄ + η +
Qa,t
Qa,t
τv + Yt i=1
t
1
R̂t
Qa,t−1 v̄ + η + ξa,t−1 − v̄ +
τv + Yt i=1
t
1
qa,t ea,t + Qa,t η −
Yt η +
(Ai − Ai−1 )τui ui
τv + Yt
i=1
t−1
1
Qa,t−1 η −
(Ai − Ai−1 )τui ui
Yt−1 η +
τv + Yt−1
i=1
t
Qa,t
(Ai − Ai−1 )τui ui
qa,t ea,t +
τv η −
τv + Yt
i=1
∼N(0,qt )
−
Qa,t−1
τv + Yt−1
τv η −
=
qa,t ea,t −
t−1
(Ai − Ai−1 )τui ui
i=1
Qa,t
Qa,t−1
−
τv + Yt
τv + Yt−1
∼N(0,qt )
=
qa,t ea,t −
Qa,t
Qa,t−1
−
τv + Yt
τv + Yt−1
τv η −
(Ai − Ai−1 )τui ui
i=1
t−1
−
Qa,t
(At − At−1 )τui ut
τv + Yt ∼N(0,yt )
∼N(0,τv +Yt−1 )
ft−1 −
Qa,t
gt ,
τv + Yt
where f1 = τv η ∼ N (0, τv ), ft = ft−1 + gt , gt ∼ N (0, yt ), ft |ft−1 ∼ N (ft−1 , yt ).
We note that the coefficient in front of ft−1 is strictly positive. The total trading volume at times t and t + 1 then
have the form:
Wt
=
|α1a,t ξa,t − α2a,t ft−1 − α3a,t gt |,
a
Wt+1
=
|α1a,t+1 ξa,t+1 − α2a,t+1 (ft−1 + gt+1 ) − α3a,t+1 gt+1 |,
a
where ξa,t , ξa,t+1 , ft−1 , gt , and gt+1 are all jointly independent, and all α’s are positive.
The result now follows from the following lemma:
50
Lemma 18 Assume that Ã, B̃ and z̃ are independent random variables with standardized normal distributions. Then,
for any a > 0 and b > 0, Cov(|aà + z̃|, |bB̃ + z̃|) > 0.
Proof of Lemma 18: First, we note that if Cov(X̃, Ỹ |Z̃1 , · · · , Z̃n ) > 0 for all realizations of the random variables Z̃1 , . . . , Z̃n ,
then it must be that Cov(X̃ , Ỹ ) > 0 unconditionally. This follows from the law of iterated expectations, since
Cov(X̃ , Ỹ ) = E[X̃ Ỹ ] − E[X̃]E[Ỹ ] = E E[X̃ Ỹ ] − E[X̃]E[Ỹ ]|Z̃ = E[Cov(X̃, Ỹ )|Z̃] > 0.
Now, define Z1 = a|Ã|, Z2 = b|B̃|. Then it follows that E[|aà + z̃| |Z1 ] = E[max(Z1 , |z̃|)], E[|bB̃ + z̃| |Z2 ] =
E[max(Z2 , |z̃|)], and E[|aà + z̃||bB̃ + z̃| |Z1 , Z2 ] = E[max(Z1 , |z̃|) max(Z1 , |z̃|)], so
Cov(|aà + z̃|, |bB̃ + z̃| |Z1 , Z2 ) = Cov(max(Z1 , |z̃|), max(Z2 , |z̃|) |Z1 , Z2 ) > 0
where the inequality follows from max(c, x) being a monotone transformation of x, so max(Z1 , |z̃|), and max(Z2 , |z̃|) are
comonotonic. Given the argument, the positivity must then also hold unconditionally, and the lemma therefore follows.
Now, this lemma implies that all the terms that make up Wt and Wt+1 have pairwise positive covariances, and it is
indeed the case that Cov(Wt , Wt+1 ) > 0, showing the second result.
For the third result, we proceed as follows: It follows from (75) that
v − pt =
t
1
1
ft = η −
R̂i .
τv + Yt
τv + Yt i=1
(76)
Therefore, since v − pt−1 is independent of pt−1 , the same holds for ft−1 (and, of course for gt ), trading volume is indeed
independent of pt−1 , and further of pt−i for all i > 1, showing the third result.
For the first result, from (76), the relationship
|pt − pt−1 | = 1
1
ft −
ft−1 =
τv + Yt−1
τv + Yt
1
1
1
ft−1 −
−
gt τ +Y
τ
+
Y
τ
+
Y
v
v
t
v
t
t−1
follows, and a similar argument as for trading volume over time then implies that Cov(|pt − pt−1 |, Wt ) > 0, showing the
first result.
We are done.
51
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